Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 25, 2000, 3–30

GROWTH OF OPERATOR VALUED MEROMORPHIC FUNCTIONS

Olavi Nevanlinna

Helsinki University of Technology, Institute of Mathematics P.O. Box 1100, FIN-02015 HUT, Finland; Olavi.Nevanlinna@hut.fi

Abstract. We discuss two characteristic functions, T_{∞} and T_{1} , to measure the growth of operator valued meromorphic functions. The smaller one, T_{∞}, is based on the operator norm, while T_{1} works within finitely trace class meromorphic functions and is preserved in inversion. Applications to perturbation analysis of linear operators are given.

1. Introduction

If f denotes a meromorphic function satisfying a normalization condition f(0) = 1, then the first main theorem in the value distribution theory, [14], [15], says in particular that

(1.1)  T (r, f) = T (r, \frac{1}{f} ).

Here T is the Nevanlinna characteristic function which measures the size of f in the disc |z| ≤ r. The theory has since 1925 been generalized in many ways but not to functions taking values in a space of linear operators, where, after all, an inversion is available. The obvious way of generalizing the characteristic function T to vector valued meromorphic functions goes by replacing the absolute value by the norm. This leads to a characteristic function, let us call it T_{∞}, which codes in a natural way many properties of such meromorphic functions but supports no analogy to (1.1). In fact, if for example our meromorphic function is simply I −zA where I is the identity and A is a bounded operator, then knowing the growth of T_{∞}(r, I − zA) does not tell us how its inverse (I − zA)^{−1} grows as a meromorphic function—or whether it is meromorphic at all.

In numerical analysis we meet questions of the following nature. If A is a bounded linear operator in a Hilbert space, how small can ∥p(A)∥ be if p is a monic polynomial of given degree, or how fast this decays with the degree, see [9], [11]. The usual tool is the holomorphic functional calculus in which you would treat the resolvent as an analytic function outside the spectrum and write

(1.2)  p(A) = \frac{1}{2πi} ∫_{Γ} p(λ)(λI − A)^{−1} dλ.

1991 Mathematics Subject Classification: Primary 30G30, 47A10, 47B10.

Here Γ would surround the spectrum within a suitable distance so that the resolvent would be of moderate size along it. However, often the answers to this type of questions are not sensitive to low rank perturbations but this we cannot conclude directly from (1.2) as the spectrum, (and hence a suitable path Γ) can be a rather wildly behaving function in low rank perturbations. Think now the resolvent as a meromorphic function rather than an analytic one outside the spectrum. Multiply it by a function χ(λ) = 1 + a_{1}λ^{−1} + a_{2}λ^{−2} + ⋯, vanishing at the poles such that the product χ(λ)(λI − A)^{−1} is entire in the variable 1/λ. We show in this paper that the growth of T_{∞} as |λ| → 0 of the resolvent is insensitive in low rank perturbations. Knowing the growth allows one to estimate ∥χ(λ)(λI −A)^{−1}∥ without knowledge of the actual locations of possible poles. Finally, instead of (1.2) we would use the fact that the coefficients p_{k}(A) in the expansion

χ(λ)(λI − A)^{−1} = ∑ p_{k}(A)λ^{−1−k}

satisfy

p_{k}(A) = \frac{1}{2πi} ∫_{|λ|=r} λ^{k}χ(λ)(λI − A)^{−1} dλ

and therefore they decay with a speed determined by the growth of χ(λ)(λI−A)^{−1} .

Another place where it is useful to think the resolvent as a meromorphic function is in estimating the powers of an operator. There are well-known results in finite dimensional spaces of the form that provided a resolvent growth condition outside the unit circle is satisfied then the powers are bounded by a constant which depends on the dimension. In finite dimensional spaces resolvents are rational functions and the dimension gives an upper bound for their degree. If we measure the resolvent as a meromorphic function using T_{∞} we obtain dimension free results in Hilbert spaces, see [10], [13].

The full power of the first main theorem is obtained by applying (1.1) to f −a, with a complex constant a. After all, f and f −a are “large” at same places. This gave a quantitative meaning for a value to be exceptional in the Picard sense. Now, a simple analogy of this with operators is the following, see Examples 5.2.7, 5.2.8 in [9], and Example 4.3 below. A solution operator to a second order differential equation with pure initial value conditions is a quasinilpotent Volterra operator while the “same” differential equation with two point boundary conditions gives a self-adjoint Fredholm operator. The analogy with initial conditions corresponding the exceptional case is obvious. Observe that the change of boundary conditions is just a rank-1 perturbation.

In [12] we proposed a characteristic function, let us call it here T_{1} , for square matrices with meromorphic elements which satisfies the analogy of (1.1):

(1.3)  T_{1}(r, F) = T_{1}(r, F^{−1})

provided F (0) = I. The starting point was the following observation: writing

log | det F | = ∑ log^{+} σ_{j}(F) − ∑ log^{+} σ_{j}(F^{−1})

where σ_{j} denote the singular values, provides the analogy of the key formula log |f| = log^{+} |f| − log^{+}(1/|f|).

In this paper we extend T_{1} for functions of the form I −F where F is finitely 𝒮_{1} -meromorphic, that is, functions which take values in the trace class with the extra property that the principal parts of poles are of finite rank each. Now, these have inverses (I − F)^{−1} in the same class and the analogy of (1.3) holds.

We can use T_{1} in perturbation analysis as follows. Suppose F^{−1} and G are meromorphic, then

T_{∞}(r, (F + G)^{−1}) ≤ T_{∞}(r, F^{−1}) + T_{∞}(r, (I + F^{−1}G)^{−1}).

If now F and G are such that F^{−1}G is finitely 𝒮_{1} -meromorphic, then

T_{∞}(r, (I + F^{−1}G)^{−1}) ≤ T_{1}(r, (I + F^{−1}G)^{−1}) = T_{1}(r, I + F^{−1}G) + C

where C depends on F^{−1}G at the origin.

The basic concepts are presented in Section 2, Section 3 contains the identity for inversion, while in Section 4 we present some applications to perturbation theory. Finally, the concepts are applied to the resolvent in Section 5.

2. Basic definitions

In this paper H is a separable complex infinite dimensional Hilbert space. We discuss two different extensions of the theory of scalar meromorphic functions to operator valued functions. The first one reduces to the scalar theory when applied to operators of the form fI where f is a meromorphic scalar function and I the identity operator. The second one agrees with the scalar theory when applied to operators of the form diag(f, 1, 1, 1, …).

Let

(2.1)  F: z ↦ F (z) = ∑_{−m}^{∞} A_{j}(z − z_{0})^{j}

be such that it has expansions of this form around every z_{0} in |z_{0}| < R ≤ ∞. Here A_{j} are bounded linear operators in H and ∑_{−m}^{∞} ∥A_{j}∥η^{j} < ∞ for some η > 0, and A_{−m} is nontrivial.

Definition 2.1. Let F be a meromorphic operator valued function as above. If −m < 0, then F has a pole at z_{0} of order m, otherwise F is analytic at z_{0} . Denote m(z_{0}) := max{m, 0} and define

(2.2)  n_{∞}(r, F) := ∑_{|b|≤r} m(b).

Thus n_{∞} counts the poles together with their orders. We then define as usual

(2.3)  N_{∞}(r, F) := ∫_{0}^{r} \frac{n_{∞}(t, F) − n_{∞}(0, F)}{t} dt + n_{∞}(0, F) log r.

Also, we set

(2.4)  m_{∞}(r, F) := \frac{1}{2π} ∫_{−π}^{π} log^{+} ∥F (re^{iϕ})∥ dϕ

and

(2.5)  T_{∞}(r, F) := m_{∞}(r, F) + N_{∞}(r, F).

Finally, we denote

(2.6)  M_{∞}(r, F) := sup_{|z|=r} ∥F (e^{iϕ})∥.

We collect the main properties of T_{∞} into the following two theorems.

Theorem 2.1. Let F, G and F_{i} be meromorphic for |z| < R ≤ ∞. Then T_{∞}(r, F) is a nonnegative and nondecreasing function in r for 0 ≤ r < R which is convex in the variable log r. The following inequalities hold

(2.7)  T_{∞}(r, F G) ≤ T_{∞}(r, F) + T_{∞}(r, G),
(2.8)  T_{∞}(r, ∑_{1}^{k} F_{i}) ≤ ∑_{1}^{k} T_{∞}(r, F_{i}) + log k.

Proof. The proof is as for the scalar case. See any recent text book on the topic, or compare with the proof of Theorem 2.4. The important fact is that log^{+} ∥F ∥ is subharmonic whenever F is analytic [1].

Theorem 2.2. If F is analytic for |z| < R and 0 < r < θr < R, then

(2.9)  T_{∞}(r, F) ≤ log^{+} M_{∞}(r, F) ≤ \frac{θ + 1}{θ − 1}T_{∞}(θr, F).

Proof. Again, since log^{+} ∥F ∥ is subharmonic if F is analytic, the claim follows using the Poisson–Jensen formula as in the scalar case.

If f is a scalar meromorphic function (with f(0) = 1) then by the first main theorem

T (r, f) = T (r, \frac{1}{f} ).

It is clear that we cannot have an identity in the operator valued case for T_{∞}. For example, if

F: z ↦ I − zA

where A is a bounded operator, then T_{∞}(r, F) = log^{+} r + 𝒪(1) but the inverse (I − zA)^{−1} need not even be meromorphic in the whole plane.

Example 2.1. Let in particular H := l_{2} and A := diag(α_{j}) where {α_{j}} is a decreasing sequence of positive numbers. Now (I − zA)^{−1} has poles at 1/α_{j} and it is meromorphic in the whole plane if and only if lim α_{j} = 0. That is, if and only if A is compact. Observe that

m_{∞}(r, (I − zA)^{−1}) = 𝒪(1)

no matter whether A is compact. All growth in T_{∞}(r, (I − zA)^{−1}) comes from counting the poles (this is true for all self-adjoint operators, see Theorem 5.5). In fact

T_{∞}(r, (I − zA)^{−1}) = ∑ log^{+}(α_{j}r) + 𝒪(1).

Observe that here the eigenvalues α_{j} are also the singular values of A. If we like to conclude from the growth of I − zA the growth for its inverse we have to see more growth in it. We do this by including all the singular values into the characteristic function, and not just the largest one, the norm.

Definition 2.2. The singular values σ_{j}(B) of a bounded operator B are

(2.10)  σ_{j}(B) := inf_{rank(B_{j})<j} ∥B − B_{j}∥.

Also, we set

(2.11)  σ_{∞}(B) := lim_{j→∞} σ_{j}(B).

Remark 2.1. It is well known that K is compact if and only if σ_{∞}(K) = 0.

Lemma 2.1. If K is compact, then

(2.12)  σ_{∞}(I − K) = 1

and in particular σ_{j}(I − K) ≥ 1 for all j.

Proof. Since K is compact, there exists a sequence of finite rank operators, {A_{j}} converging to K and thus

σ_{∞}(I − K) = lim_{i→∞} σ_{i}(I − K) ≤ lim inf ∥I − K + A_{j}∥ ≤ 1.

If the inequality would be proper, then for some k, σ_{k}(I − K) < 1 and further, there would be an operator K_{k} of rank less than k such that

∥I − K − K_{k}∥ < 1.

Taking large enough j we would then have

∥I − A_{j} − K_{k}∥ + ∥K − A_{j}∥ < 1.

But A_{j}+K_{k} is finite dimensional and its distance to identity cannot be less than 1.

Definition 2.3. A compact operator K is in the Schatten class 𝒮_{p}, if

∥K∥_{p} := (∑_{j} σ_{j}(K)^{p})^{1/p} < ∞.

Lemma 2.2. If K ∈ 𝒮_{1} , then
(2.13)  ∑_{j} log^{+} σ_{j}(I − K) ≤ ∥K∥_{1}.

Proof. The claim follows from σ_{j}(I −K) ≤ 1+σ_{j}(K) and log^{+}(1+σ_{j}(K)) ≤ σ_{j}(K).

Definition 2.4. Let A be a bounded linear operator. We define [12] the total logarithmic size of A by

(2.14)  s(A) := ∑_{j} log^{+} σ_{j}(A)

whenever finite.

Lemma 2.3. If A and B are compact, then for all k

(2.15)  ∑_{1}^{k} |σ_{j}(I + A + B) − σ_{j}(I + A)| ≤ ∑_{1}^{k} σ_{j}(B).

Proof. Fix any k. If B is bounded one can define |||B|||_{k} := ∑_{j=1}^{k} σ_{j}(B). These are sometimes called Ky Fan norms. In finite dimensional spaces they have the following property. Let Σ(B) := diag(σ_{j}(B)) where the singular values are listed decreasingly. Then

|||Σ(A) − Σ(B)|||_{k} ≤ |||A − B|||_{k},

see [6, p. 448]. So, (2.15) holds in finite dimensional spaces. Since A and B are compact we can approximate them within any ε > 0 by finite rank operators A_{m} and B_{n}. But then we can take a large enough finite dimensional subspace H_{0} such that A_{m} and B_{n} are invariant in it, and vanish in the orthogonal complement. We can then add extra dimensions to it, so many as needed to guarantee that the k largest singular values of I + A_{m} and of I + A_{m} + B_{n}, when restricted to that subspace, are all ≥ 1. This is possible by Lemma 2.1. But then the result in finite dimensional spaces can be used and we have

\begin{aligned}∑_{1}^{k} |σ_{j}(I + A + B) − σ_{j}(I + A)| & ≤ ∑_{1}^{k} |σ_{j}(I + A_{m} + B_{n}) − σ_{j}(I + A_{m})| + 3kε \\ & ≤ ∑_{1}^{k} σ_{j}(B) + 4kε. \end{aligned}

Continuity Lemma 2.4. If A, B ∈ 𝒮_{1} , then

(2.16)  |s(I + A) − s(I + B)| ≤ ∥A − B∥_{1}.

Proof. If a and b are nonegative numbers then

| log(1 + a) − log(1 + b)| ≤ |a − b|.

But σ_{j}(I + A), σ_{j}(I + B) ≥ 1 so that

| log^{+} σ_{j}(I + A) − log^{+} σ_{j}(I + B)| ≤ |σ_{j}(I + A) − σ_{j}(I + B)|.

The conclusion now follows from Lemma 2.3.

Theorem 2.3. Suppose F: z ↦ F (z) is an analytic 𝒮_{1} -valued function for |z − z_{0}| < R_{0} . Then the function u:

z ↦ u(z) := s(I − F (z)) = ∑_{j} log^{+} σ_{j}(I − F (z))

is continuous and subharmonic for |z − z_{0}| < R_{0} .

To prove this we first state some lemmas.

Approximation Lemma 2.5. If F is an analytic 𝒮_{1} -valued function in |z − z_{0}| < R_{0} , then F can be approximated in 𝒮_{1} by finite rank polynomials uniformly in discs |z − z_{0}| ≤ η < R_{0} .

Proof. Let the Taylor coefficients of F at z_{0} be A_{j}. Then in particular

∑_{n+1}^{∞} ∥A_{j}∥_{1}η^{j} → 0 as n → ∞.

Let A_{j}^{(n)} be a finite rank approximation to A_{j} such that

∥A_{j} − A_{j}^{(n)}∥_{1} < \frac{1}{n} \frac{1}{j!}.

Set

F_{n}(z) := ∑_{j=0}^{n} A_{j}^{(n)}(z − z_{0})^{j}.

Then for |z − z_{0}| < η

∥F (z) − F_{n}(z)∥_{1} ≤ ∑_{j=0}^{n} \frac{1}{n} \frac{1}{j!}η^{j} + ∑_{n+1}^{∞} ∥A_{j}∥_{1}η^{j} ≤ \frac{1}{n}e^{η} + ∑_{n+1}^{∞} ∥A_{j}∥_{1}η^{j} → 0

as n → ∞.

Lemma 2.6. The function ∑ log^{+} σ_{j}(F (z)) is subharmonic for analytic F in finite dimensional spaces.

Proof. This is in [2].

Proof of Theorem 2.3. We start by approximating F by a finite rank polynomial F_{n} in a neighborhood of a point z_{0} . Then there exists a subspace, say H_{n} of dimension d ≤ 2 rank(F_{n}) such that F_{n} is invariant in H_{n} and vanishes in the orthogonal complement. Now, at most d singular values of I − F_{n}(z) can be different from 1. Denote by A_{n}(z) the finite dimensional operator obtained by restricting I − F_{n}(z) to H_{n}. Then by Lemma 2.6

z ↦ ∑ log^{+} σ_{j}(A_{n}(z))

is subharmonic. It is continuous as A_{n} is a polynomial and the singular values are continuous. By construction, however

∑ log^{+} σ_{j}(A_{n}(z)) = ∑ log^{+} σ_{j}(I − F_{n}(z)).

But combining the Approximation and Continuity lemmas we conclude that

z ↦ ∑ log^{+} σ_{j}(I − F (z)) = s(I − F (z))

is the uniform limit of subharmonic continuous functions and therefore itself also subharmonic and continuous.

Theorem 2.3 suggests that we could look at the following quantities.

Definition 2.5. If F is an analytic 𝒮_{1} -valued function then we set

(2.17)  M_{1}(r, I − F) := sup_{|z|=r} ∏_{1}^{∞} σ_{j}(I − F (z))

and

(2.18)  m_{1}(r, I − F) := \frac{1}{2π} ∫_{−π}^{π} s(I − F (re^{iϕ})) dϕ.

Recall, that in (2.17) σ_{j}(I − F (z)) ≥ 1 by Lemma 2.1.

Example 2.2. An operator valued function can be entire in the uniform norm but have a finite domain if considered as an analytic function taking values in 𝒮_{1} . In fact, let A_{j} be a diagonal operator with positive decaying diagonal elements α_{j,k} such that

∑_{k=1}^{∞} α_{j,k} = 1

while α_{j,1} = 1/j!. Then ∥A_{j}∥ = 1/j! while ∥A_{j}∥_{1} = 1 and if F (z) := ∑_{j} A_{j}z^{j} then

∥F (r)∥_{1} = \frac{r}{1 − r}

while

∥F (r)∥ = e^{r} − 1.

For 1 + x ≤ e we have x/(e − 1) ≤ log(1 + x) ≤ x which then gives

\frac{1}{(e − 1)(1 − r)} ≤ log M_{1}(r, I + F) ≤ \frac{1}{1 − r}

while

log M_{∞}(r, I + F) = r.

What still remains, is to specify what we here mean by a pole and how to count their multiplicities. As long as F is analytic as an 𝒮_{1} -valued function, either I − F (z) has an inverse

(I − F (z))^{−1} = I + F (z)(I − F (z))^{−1}

with

F (z)(I − F (z))^{−1} ∈ 𝒮_{1},

or 1 is an eigenvalue of F (b) with a finite dimensional eigenspace. In particular, at such a point b, F (z)(I − F (z))^{−1} has an expansion of the form

∑_{−m}^{∞} B_{j}(z − b)^{j}

where the operators B_{j} are all in 𝒮_{1} and of finite rank when j < 0.

Example 2.3. Let F (z) := diag(α_{j}/(1 − z)) where {α_{j}} is a decreasing sequence of positive numbers so that their sum is finite. Then F is clearly a meromorphic 𝒮_{1} -valued function in the whole plane. However, the inverse

(I − F)^{−1} = I + diag(\frac{α_{j}}{1 − z − α_{j}} )

has an accumulation point of poles at 1 and we see that it is essential to assume the principal parts to be of finite rank.

Definition 2.6. We say that a meromorphic operator valued function F is finitely 𝒮_{1} -meromorphic in |z| < R_{1} ≤ ∞, for short F ∈ ℱ_{1}(R_{1}), if the Laurent-series (2.1) converges in 𝒮_{1} , and the coefficients A_{j} in the principal part

∑_{−m}^{−1} A_{j}(z − z_{0})^{j}

are of finite rank, at every pole z_{0} .

In [16] there are some related concepts. In particular, compact operator valued functions such that the principal parts are of finite rank are there called essentially meromorphic.

Lemma 2.7. If F, G ∈ ℱ_{1}(R), then
(2.19a)  F G ∈ ℱ_{1}(R),
(2.19b)  F + G ∈ ℱ_{1}(R),

and

(2.19c)  F (I − F)^{−1} ∈ ℱ_{1}(R).

If A is an analytic operator valued function in |z| < R, then also

(2.19d)  AF and F A ∈ ℱ_{1}(R).

Proof. All claims are easy. Recall that for linear operators A, B: ∥AB∥_{1} ≤ ∥A∥∥B∥_{1} .

Definition 2.7. If F ∈ ℱ_{1}(R) then set

(2.20)  μ(z_{0}) := lim sup_{z→z_{0}} \frac{s(I − F (z))}{log \frac{1}{|z − z_{0}|}}.

Lemma 2.8. If F ∈ ℱ_{1}(R), then at poles b, μ(b) is a positive integer, depending only on the principal part ∑_{−m}^{−1} A_{j}(z − b)^{j}.

Proof. Write

F (z) =: ∑_{−m}^{−1} A_{j}(z − b)^{j} + G(z)

with G analytic near b. Then by the Continuity Lemma

|s(I − F (z)) − s(I − F (z) + G(z))| ≤ ∥G(z)∥_{1}.

But ∥G∥_{1} is bounded near b and has thus no effect on μ(b). Now

F (z) − G(z) = ∑_{−m}^{−1} A_{j}(z − b)^{j}

is of finite rank and therefore (I −F +G) only has a finite number, say n, singular values which are not identically 1. As for any A, |σ_{j}(I − A) − σ_{j}(A)| ≤ 1, we have

|s(I − F (z) + G(z)) − s(F (z) − G(z))| ≤ n

which shows that μ(b) depends only on the principal part.

We can now assume without restricting the generality that b = 0 and that F (z) = ∑_{−m}^{−1} A_{j}(z − b)^{j} is a d × d matrix. Let λ_{j}(z) denote the eigenvalues of F (z)^{∗}F (z), ordered decreasingly. These are nonnegative and their square roots are the singular values. The characteristic polynomial can be “expanded by diagonal elements”:

det(λI − F (z)^{∗}F (z)) = λ^{d} − b_{1}(z)λ^{d−1} + ⋯ + (−1)^{d}b_{d}(z)

where b_{1} = ∑ λ_{j}, b_{2} = ∑_{i≠j} λ_{i}λ_{j} etc. As all eigenvalues are nonnegative, the functions b_{j} are nonnegative as well. Also, if b_{k} = 0 then b_{j} = 0 for j = k + 1, …, d. The coefficients b_{j} are sums of all principal minors of order j in det(F^{∗}F). These are determinants of j × j submatrices which in turn are of the form F_{j}^{∗}F_{j} where each F_{j} is a d × j matrix consisting of j columns of F. Let I_{j} denote a selection of j rows from a matrix so that F_{j}(I_{j}) denotes a j × j submatrix of F_{j}. The Cauchy–Binet Theorem allows us to conclude that then

b_{j} = ∑ | det F_{j}(I_{j})|^{2}

where the summation is over all j × j minors F_{j}(I_{j}) of F. But determinants are meromorphic and therefore there exists c_{j} > 0 and an integer m_{j} such that

b_{j}(z) = c_{j}(1 + o(1))r^{2m_{j}}

as |z| = r → 0. Consider now b_{1} = ∑_{1}^{d} λ_{j}. As the eigenvalues are numbered decreasingly we have

\frac{1}{d}b_{1} ≤ λ_{1} ≤ b_{1}

which further implies

\frac{c_{1}}{d} ≤ lim inf_{z→0} \frac{λ_{1}(z)}{r^{2m_{1}}} ≤ lim sup_{z→0} \frac{λ_{1}(z)}{r^{2m_{1}}} ≤ c_{1}.

For the coefficient b_{2} we have in the same way

λ_{1}λ_{2} ≤ b_{2} ≤ \binom{d}{2}λ_{1}λ_{2}.

This implies

\frac{c_{2}}{c_{1}\binom{d}{2}} ≤ lim inf_{z→0} \frac{λ_{2}(z)}{r^{2(m_{2}−m_{1})}} ≤ lim sup_{z→0} \frac{λ_{2}(z)}{r^{2(m_{2}−m_{1})}} ≤ \frac{c_{2}d}{c_{1}}.

Continuing this way we see that if λ_{j} is not identically 0, then there exists constants a_{j} > 0 and an integer k_{j} such that

a_{j}^{2} ≤ lim inf_{z→0} \frac{λ_{j}(z)}{r^{2k_{j}}} ≤ lim sup_{z→0} \frac{λ_{j}(z)}{r^{2k_{j}}} ≤ \frac{1}{a_{j}^{2}} .

Taking the logarithm and dividing by 2 gives

log a_{j} ≤ lim inf_{z→0}[log σ_{j}(F (z)) + k_{j} log \frac{1}{r}]

≤ lim sup_{z→0}[log σ_{j}(F (z)) + k_{j} log \frac{1}{r}] ≤ log \frac{1}{a_{j}} .

Since the eigenvalues were ordered decreasingly there is a J such that k_{j} < 0 for j ≤ J. Summing over j then gives

α ≤ lim inf_{z→0}[∑ log^{+} σ_{j}(F (z)) + ∑_{j=1}^{J} k_{j} log \frac{1}{r}]

≤ lim sup_{z→0}[∑ log^{+} σ_{j}(F (z)) + ∑_{j=1}^{J} k_{j} log \frac{1}{r}] ≤ β

where α := ∑_{j=1}^{J} log a_{j} and β := ∑_{j=1}^{J} log(1/a_{j}). Thus, in particular, μ(0) := − ∑_{j=1}^{J} k_{j} is an integer.

Lemma 2.9. If F ∈ ℱ_{1}(R), then
(2.21)  μ(z_{0}) = lim_{z→z_{0}} \frac{s(I − F (z))}{log \frac{1}{|z − z_{0}|}},

and there are constants α and β such that

α ≤ lim inf_{z→z_{0}}[s(I − F (z)) − μ(z_{0}) log \frac{1}{|z − z_{0}|}]

≤ lim sup_{z→z_{0}}[s(I − F (z)) − μ(z_{0}) log \frac{1}{|z − z_{0}|}] ≤ β.

Proof. These inequalities were actually derived while proving the previous lemma. The limit in (2.21) is then obtained by dividing with log(1/(|z − z_{0}|)).

Definition 2.8. If F ∈ ℱ_{1}(R_{1}), then for |z| ≤ R < R_{1} set (when z is not a pole)

(2.22)  u_{R}(z) := s(I − F (z)) + ∑_{|b|≤R} μ(b) log|\frac{R(z − b)}{R^{2} − \overline{b}z}|.

At poles b define u_{R}(b) := lim sup_{z→b} u_{R}(z).

Lemma 2.10. u_{R} is subharmonic in |z| ≤ R and equals s(I −F) on |z| = R. Proof. By Lemma 2.9 u_{R} is bounded near poles and as it is a sum of subhar-

monic and harmonic functions except at poles we conclude that u_{R} is subharmonic also at poles.

It is now natural to count the multiplicities as follows. Definition 2.9. If F ∈ ℱ_{1}(R_{1}), then for r < R_{1} denote

(2.23)  n_{1}(r, I − F) := ∑_{|b|≤r} μ(b).

Likewise,

(2.24) N_{1}(r, I − F) := ∫_{0}^{r} \frac{n_{1}(t, I − F) − n_{1}(0, I − F)}{t} dt + n_{1}(0, I − F) log r.

As m_{1} was given already in Definition 2.5 we can finally set

T_{1}(r, I − F) := m_{1}(r, I − F) + N_{1}(r, I − F).

Lemma 2.11. If F ∈ ℱ_{1}(R_{1}), then in the notation above, for r ≤ R < R_{1}

(2.25)  \frac{1}{2π} ∫_{−π}^{π} u_{R}(re^{iϕ}) dϕ = T_{1}(r, I − F) − N_{1}(R, I − F).

Proof. This is a direct calculation, based on

\frac{1}{2π} ∫_{−π}^{π} log |a − e^{iϕ}| dϕ = log^{+} |a|.

The following theorem summarizes the main properties of the characteristic function T_{1} .

Theorem 2.4. If F is finitely 𝒮_{1} -meromorphic in |z| < R_{1} ≤ ∞, then T_{1}(r, I − F) is well defined, nonnegative, nondecreasing in r < R_{1} such that it is convex as a function of log r. It satisfies

T_{∞}(r, I − F) ≤ T_{1}(r, I − F).

If G is another function in ℱ_{1}(R_{1}), then

(2.26)  T_{1}(r, (I − F)(I − G)) ≤ T_{1}(r, I − F) + T_{1}(r, I − G).

Proof. Positivity of T_{1} is clear from the definition. It is increasing and convex in the variable log r by Lemma 2.11 as this is a general fact of mean values of subharmonic functions, see [5, p. 127]. The inequality (2.26) follows from the next lemma.

Lemma 2.12. If A, B ∈ 𝒮_{1} , then

(2.27)  s((I − A)(I − B)) ≤ s(I − A) + s(I − B).

Proof. It is clear from the previous proofs that this inequality again follows by approximation arguments if we have for all d × d-matrices A, B

s(AB) ≤ s(A) + s(B).

However, this follows easily from the following inequality of A. Horn, see [7, Theorem 3.3.4]: for all k ≤ d

∏_{1}^{k} σ_{j}(AB) ≤ ∏_{1}^{k} σ_{j}(A)σ_{j}(B).

The analogue of Theorem 2.2 holds also for T_{1} , by the same argument, since s(I − F) is subharmonic.

Theorem 2.5. If F is an analytic 𝒮_{1} -function for |z| < R_{1} and 0 < r < θr < R_{1} , then

(2.28)  T_{1}(r, I − F) ≤ log^{+} M_{1}(r, I − F) ≤ \frac{θ + 1}{θ − 1}T_{1}(θr, I − F).

3. An identity for inversion

We shall establish the identity starting from the corresponding identity for the determinant function.

Lemma 3.1. Let f be a meromorphic scalar function in |z| < R. Let {a_{j}} denote the zeros and {b_{j}} the poles of f (only nonzero ones and repeated if multiple). Assume that at the origin

(3.1)  f(z) = ∑_{j=−ν}^{∞} c_{j}z^{j},

where c_{−ν} is the first nonvanishing coefficient. Then for r < R

(3.2) log |c_{−ν}| = \frac{1}{2π} ∫_{−π}^{π} log |f(re^{iϕ})| dϕ − ∑ log^{+} \frac{r}{|a_{j}|} + ∑ log^{+} \frac{r}{|b_{j}|} + ν log r.

Proof. This is the standard starting point in proving the first main theorem, see [15].

Lemma 3.2. If F ∈ ℱ_{1}(R), then det(I − F (z)) is meromorphic for |z| < R and for z not a pole of F, nor a zero of det(I − F (z)) we have

(3.3)  log|det(I − F (z))| = s(I − F (z)) − s((I − F (z))^{−1}).

Proof. Let F be given and fix η < R. Then there are only a finite number of poles b_{j} such that |b_{j}| ≤ η. We can write

F = ∑ P_{j} + G

where P_{j} is the principal part of F at b_{j}, and G is analytic for r < η + δ for some δ > 0. By Approximation Lemma 2.5 there exist finite rank polynomials G_{n} approximating G in 𝒮_{1} uniformly in the disc |z| ≤ η. Set now F_{n} := ∑ P_{j} +G_{n}. As F_{n} is of finite rank, det(I − F_{n}(z)) is a rational function. Since the poles are not moving with n take small neighborhoods of poles out and then the rational functions converge uniformly to det(I − F (z)) by definition of det for operators of the form I − A with A ∈ 𝒮_{1} , see e.g. [3]. The limit function is analytic outside these small neighborhoods. Now remove the poles by multiplying the rational functions by (z − b_{j})^{ν_{j}} and conclude that the limit function det(I − F (z)) has a pole of the same multiplicity ν_{j} at b_{j}. Notice in particular that there may be poles of F which are not poles of det(I − F), i.e. ν_{j} can be 0.

To prove the identity (3.3) fix z which is not a pole of F and such that det(I − F (z)) does not vanish. Denote F (z) =: A. We want to show that if 1 is not an eigenvalue of A ∈ 𝒮_{1} , then

log | det(I − A)| = s(I − A) − s((I − A)^{−1}).

Choose a sequence {A_{n}} converging to A in 𝒮_{1} with rank(A_{n}) ≤ n. Let H_{n} be a subspace of dimension d ≤ 2n such that A_{n} is invariant in H_{n} and vanishes in the orthogonal complement. We may also assume n to be large enough so that for j ≥ n

∥(I − A_{j})^{−1}∥ ≤ 2∥(I − A)^{−1}∥.

In particular, then

∥(I − A)^{−1} − (I − A_{j})^{−1}∥_{1} ≤ 2∥(I − A)^{−1}∥^{2}∥A − A_{j}∥_{1}

shows that also the differences of the inverses converge in 𝒮_{1} . Let B_{n} denote the restriction of I − A_{n} to H_{n}. Then also, B_{n}^{−1} is the restriction of the inverse of I − A_{n} to H_{n} and the determinant satisfies det B_{n} = det(I − A_{n}). However, the singular values of B_{n} and those of I − A_{n} agree only as long as they are larger than 1. But when σ_{k}(B_{n}) < 1 we can write

σ_{k}(B_{n}) = \frac{1}{σ_{d+1−k}(B_{n}^{−1})}

and here σ_{d+1−k}(B_{n}^{−1}) is again a singular value of (I − A_{n})^{−1} . Since | det B_{n}| = ∏_{1}^{d} σ_{k}(B_{n}) we obtain

log | det(B_{n})| = ∑ log^{+} σ_{k}(B_{n}) − ∑ log^{+} σ_{k}(B_{n}^{−1})

and, written differently,

log | det(I − A_{n})| = s(I − A_{n}) − s((I − A_{n})^{−1}).

Letting n → ∞ gives now the identity as both det and s are continuous in 𝒮_{1} .

Lemma 3.2 gives immediately, with f = det(I − F)

(3.4)  \frac{1}{2π} ∫_{−π}^{π} log |f(re^{iϕ})| dϕ = m_{1}(r, I − F) − m_{1}(r, (I − F)^{−1}).

Also, (3.3) and Definition 2.9 imply that

(3.5)  N(r, f) − N(r, \frac{1}{f} ) = N_{1}(r, I − F) − N_{1}(r, (I − F)^{−1}))

where N(r, f) denotes the usual averaged counting function of f = det(I − F):

(3.6)  N(r, f) = ∑ log^{+} \frac{r}{|b_{j}|} + max\{ν, 0\} log r.

Theorem 3.1. Let F be finitely 𝒮_{1} -meromorphic in |z| < R and if around origin

det(I − F (z)) = c_{−ν}z^{−ν} + c_{−ν+1}z^{−ν+1} + ⋯

then for r < R

(3.7)  T_{1}(r, I − F) = T_{1}(r, (I − F)^{−1}) + log |c_{−ν}|.

Proof. We only have to substitute (3.4) and (3.5) into (3.2).

Remark 3.1. Notice that Lemma 2.9 and (3.3) give

(3.8)  log |c_{−ν}| = a(I − F) − a((I − F)^{−1})

where

a(I − F) := lim sup_{z→0}[s(I − F (z)) − μ(0) log \frac{1}{|z|}].

This allows one to write the identity (3.7) in a symmetric form with no reference to the determinant function:

(3.9)  T_{1}(r, I − F) − a(I − F) = T_{1}(r, (I − F)^{−1}) − a((I − F)^{−1}).

If F is finitely meromorphic in a larger Schatten class, then Theorem 3.1 can still be used to give an upper bound for the inverse.

Theorem 3.2. If m is a positive integer such that F^{m} ∈ ℱ_{1}(R), then for r < R

(3.10) T_{∞}(r, (I − F)^{−1}) ≤ T_{1}(r, I − F^{m}) + (m − 1)[T_{∞}(r, F) + log 2] − log |c_{−ν}|

where c_{−ν} is as in (3.1) with f = det(I − F^{m}).

Proof. For any bounded A with 1 not in the spectrum we have

(3.11)  (I − A)^{−1} = (I − A^{m})^{−1} ∏_{j=1}^{m−1}(I − e^{iϕ_{j}}A),

where ϕ_{j} := 2πj/m. Thus, by (2.7) we have

T_{∞}(r, (I − F)^{−1}) ≤ T_{∞}(r, (I − F^{m})^{−1}) + T_{∞}(r, ∏_{j=1}^{m−1}(I − e^{iϕ_{j}}F)).

Here, by Theorem 3.1

T_{∞}(r, (I − F^{m})^{−1}) ≤ T_{1}(r, (I − F^{m})^{−1}) = T_{1}(r, I − F^{m}) − log |c_{−ν}| while Theorem 2.1 implies

T_{∞}(r, ∏_{j=1}^{m−1}(I − e^{iϕ_{j}}F)) ≤ ∑_{j=1}^{m−1}[T_{∞}(r, F) + log 2].

The first main theorem in the scalar case is obtained from the inversion identity by noting that f and f − a are large at the same time, whatever constant a we choose. Inverting f − a results in a statement on how often f is close to a. As

T (r, f) = T (r, \frac{1}{f − a}) + 𝒪(1)

f visits a equally often, independently of the value a. In the operator valued case nothing like this can hold if the “value” is taken as an “arbitrary” operator. We formulate a version here where the function F in ℱ_{1} is perturbed by a small constant operator A. Our main use of Theorem 3.1 is in perturbation theory and that is discussed in the next chapter.

Theorem 3.3. Let F ∈ ℱ_{1}(R) and A ∈ 𝒮_{1} be given. Then
(3.12)  T_{1}(r, (I − F − A)^{−1}) = T_{1}(r, I − F) + log |c_{−ν}| + ε(r, A)

where c_{−ν} is the first nonzero coefficient in the expansion of det[(I − F − A)^{−1}] at origin and

|ε(r, A)| ≤ ∥A∥_{1}.

Proof. By the Continuity Lemma we have |m_{1}(I −F −A)−m_{1}(I −F)| ≤ ∥A∥_{1} and since F and F − A have the same poles

|T_{1}(r, I − F − A) − T_{1}(r, I − F)| = |m_{1}(I − F − A) − m_{1}(I − F)| ≤ ∥A∥_{1}.

4. Perturbation results

Our first perturbation result concerns with perturbations of meromorphic operators by finite rank ones.

Theorem 4.1. Assume that F, F^{−1} and G are meromorphic for |z| < R and rank G ≤ k. Then (F + G)^{−1} is meromorphic for |z| < R and

(4.1)  T_{∞}(r, (F + G)^{−1}) ≤ (k + 1)T_{∞}(r, F^{−1}) + kT_{∞}(r, G) + C,

where

C ≤ k log 2 − log |c_{−ν}|

and c_{−ν} is the first nonzero coefficient in the Laurent series of det(I + F^{−1}G) at the origin.

Proof. Away from poles we can write F + G = F (I + F^{−1}G) and thus

(4.2)  T_{∞}(r, (F + G)^{−1}) ≤ T_{∞}(r, F^{−1}) + T_{∞}(r, (I + F^{−1}G)^{−1}).

Here F^{−1}G ∈ ℱ_{1}(R) as G is of finite rank. This allows us to use the inversion identity:

(4.3) T_{∞}(r, (I +F^{−1}G)^{−1}) ≤ T_{1}(r, (I +F^{−1}G)^{−1}) = T_{1}(r, I +F^{−1}G)−log |c_{−ν}|.

If a, b ≥ 0 then always

log(1 + ab) ≤ log^{+} a + log^{+} b + log 2.

We use this in estimating log^{+} σ_{j}(I +F^{−1}G). For j > k, σ_{j}(I +F^{−1}G) = 1 while for j ≤ k we have σ_{j}(I + F^{−1}G) ≤ 1 + ∥F^{−1}∥∥G∥. Thus

s(I + F^{−1}G) ≤ k[log^{+} ∥F^{−1}∥ + log^{+} ∥G∥ + log 2]

and

T_{1}(r, I + F^{−1}G) ≤ k[T_{∞}(r, F^{−1}) + T_{∞}(r, G) + log 2]

from which the claim follows.

Definition 4.1. Let F be a meromorphic operator valued function in the whole plane. Then the order ω_{∞} is

(4.4)  ω_{∞} := lim sup_{r→∞} \frac{log T_{∞}(r, F)}{log r}

and if 0 < ω_{∞} < ∞ then the type τ_{∞} is

(4.5)  τ_{∞} := lim sup_{r→∞} \frac{T_{∞}(r, F)}{r^{ω_{∞}}}.

If F is in ℱ_{1}(∞), then the order ω_{1} and type τ_{1} are defined in the same way for F and for I − F.

Notice that if F is entire, then it also has order and type as an entire function based on M_{∞}(r). The order is then the same as a meromorphic function while the type can be somewhat different. In fact, an easy estimate between the types can be obtained from Theorem 2.2. This holds as such also for entire 𝒮_{1} -valued functions by Theorem 2.5.

Corollary 4.1. If in addition to the assumptions of Theorem 4.1, R = ∞ and F^{−1} and G are of finite orders, ω_{∞}(F^{−1}) and ω_{∞}(G) respectively, then (F + G)^{−1} is also of finite order and

ω_{∞}((F + G)^{−1}) = max{ω_{∞}(F^{−1}), ω_{∞}(G)}.

If F^{−1} and G are also of finite type, τ_{∞}(F^{−1}) and τ_{∞}(G), then the type of (F + G)^{−1} satisfies

τ_{∞}((F + G)^{−1}) ≤ (k + 1)τ_{∞}(F^{−1}), if ω_{∞}(F^{−1}) > ω_{∞}(G),
τ_{∞}((F + G)^{−1}) ≤ k τ_{∞}(G), if ω_{∞}(F^{−1}) < ω_{∞}(G),

and

τ_{∞}((F + G)^{−1}) ≤ (k + 1)τ_{∞}(F^{−1}) + k τ_{∞}(G), if ω_{∞}(F^{−1}) = ω_{∞}(G).

Example 4.1. Let F (z) := (1−z)I and G(z) := −z diag(β_{1}, …, β_{k}, 0, 0, …), where β_{1} > β_{2} > ⋯ > β_{k} > 0. We have poles at 1 and at 1/(1 + β_{j}) and so

N_{∞}(r, (F + G)^{−1}) = log^{+} r + ∑_{1}^{k} log^{+}((1 + β_{j})r) ≥ (k + 1) log^{+} r.

For r ≥ 2 we have T_{∞}(r, F^{−1}) = log^{+} r and thus

T_{∞}(r, (F + G)^{−1}) ≥ (k + 1)T_{∞}(r, F^{−1}) for r ≥ 2.

Thus, multiplying with k + 1 is really needed.

Example 4.2. Now we look at the term kT_{∞}(r, G). Let

F := diag(k, k − 1, k − 2, …, 2, 1, 1, 1, …)

so that T_{∞}(r, F^{−1}) = 0 for all r, while G(z) := e^{z} I_{k} where I_{k} = diag(1, …, 0, …) is the rank k projection onto the k first components. Then

T_{∞}(r, G) = \frac{r}{π} .

On the other hand, (F + G)^{−1} has poles at z = log j + 2πin for all n and for j = 1, 2, …, k. Therefore

T_{∞}(r, (F + G)^{−1}) ≥ k \frac{r}{π} + 𝒪(log^{+} r).

Again, multiplying with k is really needed.

Example 4.3. The previous examples have all been based on diagonal mappings. Here we point out that rank 1 updates can change the nature of the operator dramatically. In fact, we have two operators, V^{2} and K where V^{2} is a quasinilpotent Volterra operator while K is a self-adjoint Fredholm operator which are rank 1 perturbations of each other. Now the resolvent (see the next section) of V^{2} is entire and thus N_{∞}(r, (I − zV^{2})^{−1}) = 0 while as K is selfadjoint, m_{∞}(r, (I − zK)^{−1}) = 𝒪(1) (see Theorem 5.5). So, V^{2} corresponds to the Picard exceptional case while K is as “regular” as possible. The space is L_{2}[0, 1], and

V^{2}f(t) = ∫_{0}^{t}(t − s)f(s) ds

and

Kf(t) = ∫_{0}^{1} k(t, s)f(s) ds

where k(t, s) = k(s, t) and for 0 ≤ s ≤ t ≤ 1

k(t, s) = s(t − 1).

Here V^{2} is the solution operator for the initial value problem u′′ = f with u(0) = u′(0) = 0 while K solves the same problem with boundary conditions u(0) = u(1) = 0. Thus

Kf(t) = V^{2}f(t) − V^{2}f(1)t.

It can be shown that both resolvents are of order 1/2 and of type 2/π.

The next result relaxes the assumption on finite rank to 𝒮_{1} but assumes F^{−1} to be analytic instead.

Theorem 4.2. Assume that F is meromorphic, F^{−1} analytic, and G finitely 𝒮_{1} -meromorphic for |z| < R. Then (F + G)^{−1} is meromorphic and for r < R (4.6) T_{∞}(r, (F + G)^{−1}) ≤ T_{∞}(r, F^{−1}) + max{M_{∞}(r, F^{−1}), 1}[m̂_{1}(r, G) + N_{1}(r, G)] + C,

where

m̂_{1}(r, G) := ∑_{j=1}^{∞} m(r, 1 + σ_{j}(G)),

C ≤ − log |c_{−ν}|

and c_{−ν} is the first nonzero coefficient in the Laurent series of det(I + F^{−1}G) at the origin.

Proof. We start the proof in the same way as before but the estimation of the term T_{1}(r, I + F^{−1}G) in (4.3) is different. Let, again, a, b ≥ 0. Then we have

log(1 + ab) ≤ max{a, 1} log(1 + b).

Put for short, M := max{M_{∞}(r, F^{−1}), 1}. Then this inequality gives

log^{+} σ_{j}(I + F^{−1}G) ≤ M log(1 + σ_{j}(G))

and so

s(I + F^{−1}G) ≤ M ∑ log(1 + σ_{j}(G)).

Now the claim follows from this.

Corollary 4.2. If, in addition to the assumptions in Theorem 4.2, G is analytic for |z| < R, then for r < R

(4.7) T_{∞}(r, (F + G)^{−1}) ≤ max{M_{∞}(r, F^{−1}), 1} sup_{|z|≤r} ∥G(z)∥_{1} + T_{∞}(r, F^{−1}) + C,

where

C ≤ − log |c_{−ν}|.
Proof. We have log(1 + σ_{j}(G)) ≤ σ_{j}(G) which gives the term ∥G∥_{1} .
Example 4.4. F (z) = e^{z}I and G = diag(j^{−1−ε}). For r large one has
N_{∞}(r, (F + G)^{−1}) ≥ e^{r/(1+2ε)}

while the right hand side in (4.7) is bounded from above by 𝒪(1/ε)e^{r} +(r/π)+C. Thus, the order and type of (F + G)^{−1} are the same as those of the bound.

The previous results are natural in a setting where one of the functions is treated as large and the other one as a perturbation. If both functions are finitely 𝒮_{1} -meromorphic we can formulate a result directly for their sum without inverting as the inverse is equally large by Theorem 3.1.

24 Olavi Nevanlinna

Theorem 4.3. If both F and G are in ℱ_{1}(R), then for r < R

(4.8)  T_{1}(r, I + F + G) ≤ 2T_{1}(r, I + 2F) + 2T_{1}(r, I + 2G).

The result follows immediately from the following lemma.

Lemma 4.1. If A, B ∈ 𝒮_{1} , then

(4.9)  s(I + A + B) ≤ 2s(I + 2A) + 2s(I + 2B).

Proof of the lemma. We have for any a, b

(4.10)  log^{+} \frac{1}{2}(a + b) ≤ log^{+} a + log^{+} b.

On the other hand, writing I + A + B = \frac{1}{2}[(I + 2A) + (I + 2B)] and using the fact that singular values are obtained as distances to finite rank operators, we get

σ_{2j−1}(I + A + B) ≤ \frac{1}{2}[σ_{j}(I + 2A) + σ_{j}(I + 2B)].

As σ_{2j}(I + A + B) ≤ σ_{2j−1}(I + A + B) this and (4.10) imply (4.9).

Example 4.5. If F = G then we have trivially

T_{1}(r, I + F + G) = \frac{1}{2}[T_{1}(r, I + 2F) + T_{1}(r, I + 2G)].

5. Growth of resolvents In the following we specialize to resolvents. It is natural to write them here

in the form (I − zA)^{−1} instead of the usual (λ − A)^{−1} .

We start with a result which generalizes a similar statement for quasinilpotent trace class operators, see Theorem 2.2 in Chapter X, [3].

Theorem 5.1. Assume A ∈ 𝒮_{p}. Then (I − zA)^{−1} is of order ω_{∞} ≤ p and if ω_{∞} = p then it is of zero type.

Proof. We prove first the result in the case p ≤ 1. To that end it suffices to show that for any ε > 0 we have

T_{∞}(r, (I − zA)^{−1}) ≤ εr^{p} + 𝒪(log r).

Since A ∈ 𝒮_{p} there exists an m, large enough so that

\frac{1}{p} ∑_{m+1}^{∞} σ_{j}(A)^{p} < ε.

Then, however, we can proceed as follows:

T_{∞}(r, (I − zA)^{−1}) ≤ T_{1}(r, I − zA) ≤ log M_{1}(r, I − zA)

≤ ∑_{1}^{m} log(1 + rσ_{j}(A)) + \frac{1}{p}r^{p} ∑_{m+1}^{∞} σ_{j}(A)^{p}

≤ 𝒪(log r) + εr^{p}.

Here we used the inequality log(1 + x) ≤ x^{p}/p, valid for x > 0 and 0 < p ≤ 1.

In the general case, let k be a positive integer such that k < p ≤ k + 1. Then in particular A^{k+1} ∈ 𝒮_{1} , and in fact

∑ σ_{j}(A^{k+1})^{p/(k+1)} ≤ ∑ σ_{j}(A)^{p},

see e.g. Corollary II.4.2 in [4]. We have, see Theorem 3.2, or Theorem 5.3 below,

T_{∞}(r, (I − zA)^{−1}) ≤ T_{1}(r, I − z^{k+1}A^{k+1}) + k log(1 + r∥A∥).

Here we proceed as above and in particular use

log(1 + r^{k+1}σ_{j}(A^{k+1})) ≤ \frac{k + 1}{p}r^{p}σ_{j}(A^{k+1})^{p/(k+1)}

to split the sum at a proper place in order to have the growth again bounded by εr^{p} + 𝒪(log r).

The proof above is based on

T_{1}(r, I − z^{k+1}A^{k+1}) = T_{1}(r, (I − z^{k+1}A^{k+1})^{−1}),

valid for k + 1 ≤ p. We shall next study the behavior of k^{−1}T_{1}(r, (I − z^{k}A^{k})^{−1}) as k grows.

Lemma 5.1. Let A ∈ 𝒮_{1} and {λ_{j}} denote the spectrum, indexed so that |λ_{j}| ≥ |λ_{j+1}|, each eigenvalue repeated according to the dimension of the corresponding eigenspace. Then

(5.1)  N_{1}(r, (I − zA)^{−1}) = ∑_{j} log^{+} |λ_{j}r|.

Proof. Choose r and take the Riesz spectral projection of A including all eigenvalues which are larger than, say, 1/(r + 1) in modulus. This gives a finite rank operator, say, A_{r}. Then A − A_{r} can be approximated arbitrarily well with another finite rank operator and this shows that N_{1}(r, (I − zA)^{−1}) only depends on A_{r}, compare with the Continuity Lemma 2.4. But since this is of finite rank, it can be transformed unitarily into a finite dimensional upper triangular (that is, a sum of diagonal and nilpotent) operator and then with a similarity transformation into a form where the nilpotent part is made arbitrarily small and so N_{1} only depends on the eigenvalues. In fact, if S denotes the similarity transformation, and if B = SCS^{−1} with d = dim B, then

(5.2)  s(B) ≤ d log[∥S∥ ∥S^{−1}∥] + s(C)

and therefore the multiplicity μ(1/λ_{j}) is not affected by the similarity transformation.

Observe that the right hand side of (5.1) makes sense for all compact operators as it is always a finite sum for any fixed r. We introduce the following notation for it:

(5.3)  N(r, {λ_{j}}) := ∑_{j} log^{+} |λ_{j}r|.

Now the following holds.

Theorem 5.2. Assume A ∈ 𝒮_{p} with some p. Then

(5.4)  lim_{k→∞} \frac{1}{k} T_{1}(r, (I − z^{k}A^{k})^{−1}) = N(r, \{λ_{j}\}),

where {λ_{j}} denotes the spectrum of A with multiplicities counted according to the dimensions of the corresponding eigenspaces.

Proof. Recall that if A ∈ 𝒮_{p} then A^{k} ∈ 𝒮_{1} for k ≥ p. Then for such k T_{1}(r, (I − z^{k}A^{k})^{−1}) and T_{1}(r, I − z^{k}A^{k}) are both well defined and equal. The proof is given by several simple lemmas, some of which have some independent interest.

Lemma 5.2. If A is compact, then

(5.5)  lim_{k→∞}[σ_{j}(A^{k})]^{1/k} = |λ_{j}|.

Proof of Lemma 5.2. This is Proposition 2.d.6 in [8].

The aim is to show that

(5.6)  \frac{1}{k} m_{1}(r, I − z^{k}A^{k}) → N(r, \{λ_{j}\}).

This would imply (5.4) as

T_{1}(r, (I − z^{k}A^{k})^{−1}) = T_{1}(r, I − z^{k}A^{k})

and, trivially,

N_{1}(r, I − z^{k}A^{k}) = 0.

We shall first reduce the claim into a finite dimensional problem. Since our basic claim is about a limit with a fixed r we can without lack of generality set r = 1 in the following. Choose a small 0 < δ < 1. Then take a spectral decomposition of A = A_{1} ⊕ A_{2} as follows:

A_{2} := \frac{1}{2πi} ∫_{|λ|=1−δ} λ(λI − A)^{−1} dλ.

By the spectral radius formula we have for large enough n

∥A_{2}^{n}∥^{1/n} ≤ 1 − \frac{1}{2}δ.

Lemma 5.3. Assume that A ∈ 𝒮_{p} and ρ(A) < 1. Then we have

m_{1}(1, I − z^{k}A^{k}) → 0

as k → ∞.

Proof of Lemma 5.3. If ρ(A) < ρ < 1 then for large enough n we have ∥A^{n}∥ ≤ ρ^{n}. If also n ≥ k where k such that A^{k} ∈ 𝒮_{1} , then we can estimate as |z| = 1,

σ_{j}(I − z^{n}A^{n}) ≤ 1 + ρ^{n−k}σ_{j}(A^{k})

which shows that

s(I − z^{n}A^{n}) ≤ ρ^{n−k}∥A^{k}∥_{1}.

The claim follows.

Lemma 5.4. If A ∈ 𝒮_{1} and B is of finite rank and they operate in invariant subspaces H_{A}, H_{B} respectively with H_{A} ∩ H_{B} = {0}, then

(5.7) s(I + (A ⊕ B)) ≤ s(I + A) + s(I + B) + rank(B)[log(1 + ∥A∥) + log 2].

Proof of Lemma 5.4. If j ≤ rank(B) = d then

σ_{j}(I + (A ⊕ B)) ≤ 1 + ∥A∥ + σ_{j}(I + B)

while for j > d we have

σ_{j}(I + (A ⊕ B)) ≤ σ_{j−d}(I + A).

We obtain (5.7) by taking the logarithm and summing up.

If A = A_{1} ⊕ A_{2} as above and rank A_{1} = d then Lemma 5.4 gives

(5.8) m_{1}(1, I − z^{n}A^{n}) ≤ m_{1}(1, I − z^{n}A_{1}^{n}) + m_{1}(1, I − z^{n}A_{2}^{n}) + d[log(1 + ∥A_{2}^{n}∥) + log 2].

This follows because A^{n} = A_{1}^{n} ⊕ A_{2}^{n} allows us to apply Lemma 5.4 with −z^{n}A^{n} in place of A. By Lemma 5.3 we have m_{1}(1, I − z^{n}A_{2}^{n}) → 0 and since ∥A_{2}^{n}∥ → 0, then inequality (5.8) implies

(5.9)  lim sup \frac{1}{n}m_{1}(1, I − z^{n}A^{n}) ≤ lim sup \frac{1}{n}m_{1}(1, I − z^{n}A_{1}^{n}).

What we need still to prove is the reverse inequality

(5.10)  lim inf \frac{1}{n}m_{1}(1, I − z^{n}A_{1}^{n}) ≤ lim inf \frac{1}{n}m_{1}(1, I − z^{n}A^{n})

and that the limit exists and satisfies

(5.11)  lim \frac{1}{n}m_{1}(1, I − z^{n}A_{1}^{n}) = N(1, \{λ_{j}\}).

Consider first (5.10). Let P denote the spectral projection A_{1} = P A. Then for j ≤ d we have

σ_{j}(I + A_{1}) ≤ ∥P ∥σ_{j}(I + A)

while for j > d we have σ_{j}(I + A_{1}) = 1. Thus

s(I + A_{1}) ≤ s(I + A) + d log ∥P ∥.

Applying this to −z^{n}A^{n} in place of A gives (5.10).

To prove (5.11) observe first that by construction N_{1}(1, (I − zA_{1}))^{−1} = N(1, {λ_{j}}). And recall that we have set r = 1. For |z| = 1 we have

−1 + σ_{j}(A^{n}) ≤ σ_{j}(I − z^{n}A_{1}^{n}) ≤ 1 + σ_{j}(A^{n})

which implies, as A_{1} is of rank d,

|\frac{1}{n}m_{1}(1, I − z^{n}A^{n}) − \frac{1}{n} ∑ log^{+} σ_{j}(A^{n})| ≤ \frac{1}{n}d log 2.

By Lemma 5.2 we know that

\frac{1}{n} ∑ log^{+} σ_{j}(A^{n}) → N(1, \{λ_{j}\})

which proves (5.11). The proof of Theorem 5.2 is now completed.

Definition 5.1. We denote by ρ_{∞}(A) the smallest radius such that the resolvent (I − zA)^{−1} is meromorphic for |z| < 1/ρ_{∞}(A).

Remark 5.1. Notice that the analogous concept, say ρ_{1}(A) in ℱ_{1} is not needed as by the inversion identity everything depends on whether −zA ∈ ℱ_{1} or not, that is, whether A ∈ 𝒮_{1} .

Theorem 5.3. If A is bounded, then (I − zA)^{−1} and (I − z^{k}A^{k})^{−1} are meromorphic at the same discs: ρ_{∞}(A) = ρ_{∞}(A^{k})^{1/k} and

(5.12)  T_{∞}(r, (I − z^{k}A^{k})^{−1}) ≤ kT_{∞}(r, (I − zA)^{−1})

while

(5.13) T_{∞}(r, (I − zA)^{−1}) ≤ T_{∞}(r, (I − z^{k}A^{k})^{−1}) + (k − 1) log(1 + r∥A∥).
Proof. Write, with ϕ_{j} := 2πj/k,
(I − z^{k}A^{k}) = (I − zA)(I − e^{iϕ_{1}}zA) ⋯ (I − e^{iϕ_{k−1}}zA).

This gives (5.12). Writing it for the inverses as in (3.11) gives in the same way (5.13).

So in particular, the order is preserved while the type can change somewhat. If the operator is quasinilpotent, so that the resolvent is entire, then we can also look at the growth of the maximum and here the type is preserved as well.

Theorem 5.4. If A is bounded and ρ(A) denotes the spectral radius, then for r < 1/ρ(A) we have

(5.14)  M_{∞}(r, (I − z^{k}A^{k})^{−1}) ≤ M_{∞}(r, (I − zA)^{−1}),

and

(5.15)  M_{∞}(r, (I − zA)^{−1}) ≤ [1 + r∥A∥]^{k−1}M_{∞}(r, (I − z^{k}A^{k})^{−1}).

Proof. Here (5.15) is analogous to (5.13) while (5.14) follows from writing

(I − z^{k}A^{k})^{−1} = \frac{1}{k} ∑_{1}^{k}(I − e^{iϕ_{j}}zA)^{−1}.

For reference, it is useful to formulate the following simple fact.

Theorem 5.5. Let A be bounded and selfadjoint. Then for r < 1/ρ_{∞}(A)

(5.16)  m_{∞}(r, (I − zA)^{−1}) ≤ log 2.

Proof. Since the operator is self-adjoint, we have ∥(I − zA)^{−1}∥ = 1/d(z) where

d(re^{iϕ}) := inf_{λ∈σ(A)} |1 − zλ| ≥ inf_{λ∈R} |1 − zλ| = | sin ϕ|.

Thus

\frac{1}{2π} ∫_{−π}^{π} log^{+} \frac{1}{d(re^{iφ})} dφ ≤ \frac{1}{2π} ∫_{−π}^{π} log \frac{1}{| sin φ|} dφ = log 2.

30 Olavi Nevanlinna

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Received 19 November 1997