Georgian Mathematical Journal Volume 8 (2001), Number 3, 455–468

MD-NUMBERS AND ASYMPTOTIC MD-NUMBERS OF OPERATORS

A. CASTEJÓN, E. CORBACHO, AND V. TARIELADZE

Abstract. First, the basic properties of mean dilatation (MD-) numbers for linear operators acting from a finite-dimensional Hilbert space are investigated. Among other results, in terms of first and second order MD-numbers, a characterization of isometries is obtained and a dimension-free estimation of the p-th order MD-number by means of the first order MD-number is established. After that asymptotic MD-numbers for a continuous linear operator acting from an infinite-dimensional Hilbert space are introduced and it is shown that in the case of an infinite-dimensional domain the asymptotic p-th order MD-number, rather unexpectedly, is simply the p-th power of the asymptotic first order MD-number (Theorem 3.1).

2000 Mathematics Subject Classification: Primary: 46C05; Secondary: 47A30, 47B06. Key words and phrases: Grammian, pMD-number, MD-number, asymptotic upper MD-number, asymptotic lower MD-number.

1. Introduction

For a linear operator T from a Hilbert space H with 1 ≤ n = dim H < ∞ to an arbitrary Hilbert space Y and a natural number p ≤ n, the p-th order mean dilatation number (briefly, the pMD-number) δ_{p}(T) is defined by the equality

δ_{p}(T) = (\frac{1}{c_{n,p}} ∫_{S} ⋯ ∫_{S} G(T_{1}, …, T x_{p})ds(x_{1}) … ds(x_{p}))^{1/2},  (1.1)

where S is the unit sphere in H, s is the normalized isometrically invariant measure on it, G(x_{1}, …, x_{p}) is the Grammian of (x_{1}, …, x_{p}), and

c_{n,p} = ∫_{S} ⋯ ∫_{S} G(x_{1}, …, x_{p})ds(x_{1}) … ds(x_{p}) > 0

is a (natural) normalizing constant.

The definition of pMD-numbers was suggested in [6]. Afterwards they were considered also in [5].

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

For simplicity, the first order MD-number δ_{1}(T) will be called the MD-number of T and, for it, (1.1) gives an expression

δ_{1}(T) = ( ∫_{S} ∥T x∥^{2}ds(x))^{\frac{1}{2}} .  (1.2)

Using the equality (1.2), the MD-number δ_{1}(T) can also be defined when Y is an arbitrary Banach space and in this case it has already been used as a tool of the local theory of Banach spaces (see, e.g., [7, p.81]).

In Section 2, after recalling some known facts concerning the Grammian (Remark 2.1) which motivate the definition of pMD-numbers, their properties in the finite-dimensional case are analyzed. In particular, a concrete expression for δ_{p}(T) in terms of the eigenvalues of the operator T^{∗}T is found (Proposition 2.6) and the following characterization of isometric operators is obtained: T is an isometry if and only if δ_{1}(T) = δ_{2}(T) = 1 (Proposition 2.8). Although the equality δ_{p}(T) = δ_{1}^{p}(T) does not hold in general, the validity of the following “dimension-free” estimate

δ_{p}(T) ≤ p^{\frac{p−1}{2}} δ_{1}^{p}(T)

is proved (Lemma 2.10).

Possible infinite-dimensional extensions of pMD-numbers are introduced in Section 3. According to [6], we use the net M of all finite-dimensional subspaces of H to associate two quantities

\overline{\overline{δ}}_{p}(T) := lim sup_{M∈M} δ_{p}(T |_{M}), \underline{\underline{δ}}_{p}(T) := lim inf_{M∈M} δ_{p}(T |_{M})

to a given operator T (acting from H) and a natural number p. We call \overline{\overline{δ}}_{p}(T) the asymptotic upper pMD-number of T and \underline{\underline{δ}}_{p}(T) the asymptotic lower pMDnumber of T. When they are equal, the operator is called asymptotically pMDregular, their common value is denoted by \overline{δ}_{p}(T) and called the asymptotic pMD-number. Therefore for an asymptotic pMD-regular operator T we have

\overline{δ}_{p}(T) := lim_{M∈M} δ_{p}(T |_{M}).

When p = 1, we shall simply use the term “asymptotically MD-regular”.

The main result of the section is Theorem 3.1, which asserts that, in the case of an infinite-dimensional domain, we have the following simple relation between the higher order and the first order asymptotic MD-numbers of a given operator T:

\overline{\overline{δ}}_{p}(T) = (\overline{\overline{δ}}_{1}(T))^{p}, \underline{\underline{δ}}_{p}(T) = (\underline{\underline{δ}}_{1}(T))^{p}.

This result shows that in the case of the infinite-dimensional domain it is sufficient to study only the numbers \overline{\overline{δ}}_{1}(T), \underline{\underline{δ}}_{1}(T) and \overline{δ}_{1}(T). Section 3 is concluded by Proposition 3.2, which implies, in particular, that not any operator is asymptotically MD-regular.

Notice, finally, that the numbers \overline{\overline{δ}}_{1}(T), \underline{\underline{δ}}_{1}(T) and \overline{δ}_{1}(T) can also be defined when Y is a general Banach space. The properties of the corresponding quantities in this general setting are investigated in [2].

2. Finite Dimensional Case

Throughout this paper the considered Hilbert spaces is supposed to be defined over the same field 𝕂, real ℝ or complex ℂ numbers. To simplify the notation, for the scalar product and the norm of different Hilbert spaces we use the same symbols (·|·) and ∥ · ∥.

For the Hilbert spaces H, Y we denote by L(H, Y) the set of all continuous linear operators T : H → Y.

For a given operator T ∈ L(H, Y) we denote by – ∥T ∥ the ordinary norm of T;

– T^{∗} the (Hilbert) adjoint operator of T;

– |T | the unique self-adjoint positive square root of T^{∗}T : H → H. Moreover, for a Hilbert–Schmidt operator T we denote by ∥T ∥_{HS} the Hilbert– Schmidt norm of T.

Fix a (non-zero) Hilbert space H and a natural number p such that p ≤ dim H. The p-th order Grammian G_{p} is the scalar-valued function, defined on H^{p}, which assigns the determinant of the matrix (x_{i}|x_{j})_{i,j=1}^{p} to any (x_{1}, …, x_{p}) ∈ H^{p}.

Evidently, G_{1}(x) = ∥x∥^{2} for each x ∈ H. Some other known properties of the Grammian are given in the next remark.

Remark 2.1. Fix a natural number p > 1 with p ≤ dim H and a finite sequence (x_{1}, …, x_{p}) ∈ H^{p}.

(a) We have G_{p}(x_{1}, …, x_{p}) ≥ 0. Moreover, G_{p}(x_{1}, …, x_{p}) > 0 if and only if (x_{1}, …, x_{p}) is linearly independent.

(b) G_{p}(x_{1}, …, x_{p}) ≤ ∥x_{1}∥^{2} … ∥x_{p}∥^{2}. Moreover, if x_{1}, …, x_{p} are nonzero elements, then we have G_{p}(x_{1}, …, x_{p}) = ∥x_{1}∥^{2} … ∥x_{p}∥^{2} iff x_{1}, …, x_{p} are pairwise orthogonal.

(c) Let P_{x_{1},...,x_{p}} := { ∑_{i=1}^{p} α_{i}x_{i} : 0 ≤ α_{i} ≤ 1, i = 1, …, p} be the parallelepiped

generated by the sequence (x_{1}, …, x_{p}). Then G(x_{1}, …, x_{p}) = vol^{2}(P_{x_{1},...,x_{p}}), where vol(P_{x_{1},...,x_{p}}) stands for the (p-dimensional) volume of P_{x_{1},...,x_{p}} (i.e., the Lebesgue measure of P_{x_{1},...,x_{p}} in a p-dimensional vector subspace containing (x_{1}, …, x_{p}). For this reason the quantity g_{p}(x_{1}, …, x_{p}) := G_{p}^{1/2}(x_{1}, …, x_{p}) is often simply called “the hypervolume determined by the vectors (x_{1}, …, x_{p}).”

(d) Let [x_{1}, …, x_{p−1}] be the linear span of {x_{1}, …, x_{p−1}}. Then

g_{p}(x_{1}, …, x_{p}) = g_{p−1}(x_{1}, …, x_{p−1}) · dist(x_{p}, [x_{1}, …, x_{p−1}]).

(e) If Y is another Hilbert space and T : H → Y is a continuous linear operator, then g_{p}(T x_{1}, …, T x_{p}) ≤ ∥T ∥^{p}g_{p}(x_{1}, …, x_{p}) (this is evident when p = 1, the rest follows from this and from (d) by induction, as dist(T x_{p}, [T x_{1}, …, T x_{p−1}]) ≤ ∥T ∥ dist(x_{p}, [x_{1}, …, x_{p−1}]).

(f) In the notation of the previous item suppose that dim H = p. Then g_{p}(T x_{1}, …, T x_{p}) = det(|T |)g_{p}(x_{1}, …, x_{p}). It follows that if dim H = p and (x′_{1}, …, x′_{p}), (x_{1}, …, x_{p}) are algebraic bases of H, then

\frac{g_{p}(T x′_{1}, …, T x′_{p})}{g_{p}(x′_{1}, …, x′_{p})} = \frac{g_{p}(T x_{1}, …, T x_{p})}{g_{p}(x_{1}, …, x_{p})} ,

i.e., the “volume ratio” is independent of a particular choice of basis of H and equals to det(|T |).

For a Hilbert space with dim H ≥ 1, its unit sphere with center at the origin is denoted by S_{H} or, simply, by S. When n = dim H < ∞, the uniform distribution on S, i.e. the unique isometrically invariant probability measure given on the Borel σ-algebra of S, is denoted by s. The following key equality follows directly from the isometric invariance of s:

∫_{S} \overline{(x|h_{1})}(x|h_{2})ds(x) = \frac{1}{n} (h_{1}|h_{2}), ∀h_{1}, h_{2} ∈ H,  (2.1)

where ‘bar’ stands for complex conjugation.

In the rest of this section H will be a finite-dimensional Hilbert space with dim H = n ≥ 1, I will stand for the identity operator acting in H and Y will be another Hilbert space (not necessarily finite-dimensional).

Fix a number p ∈ {1, …, n} and introduce the functional D_{p} : L(H, Y) → ℝ_{+} defined by the equality

D_{p}(T) = ∫_{S} ⋯ ∫_{S} G_{p}(T x_{1}, …, T x_{p})ds(x_{1}) … ds(x_{p}), T ∈ L(H, Y). (2.2)

Therefore, for given T the number D_{p}(T) can be viewed as the average of the squares of the volumes of the family of parallelepipeds

{T (P_{x_{1},...,x_{p}}) : (x_{1}, …, x_{p}) ∈ S × ⋯ × S}

with respect to the product s × ⋯ × s of the uniform distributions.^{1} Evidently, for p = 1 we get

D_{1}(T) = ∫_{S} ∥T x∥^{2}ds(x), ∀T ∈ L(H, Y).  (2.3)

To make easier further references, we formalize some other easy observations in the next statement.

1The choice of s seems to be natural, since it gives no preference to any of the directions (x_{1}, …, x_{p}) ∈ S × ⋯ × S. Formally, the same definition can be given, taking any probability measure μ on S instead of s, but then the properties of D_{p}(T) will depend on (the correlation operator of) μ.

Proposition 2.2. Let T ∈ L(H, Y).

D_{p}(T) = D_{p}(|T |);  (2.4)
D_{p}(T) = 0 ⟺ dim T (H) < p;  (2.5)
D_{p}(T) ≤ D_{1}^{p}(T);  (2.6)
D_{2}(T) = D_{1}^{2}(T) − \frac{1}{n}D_{1}(T^{∗}T), in particular, D_{2}(I) = 1 − \frac{1}{n}; (2.7)
D_{1}(T^{∗}T − I) = D_{1}(T^{∗}T) − 2D_{1}(T) + 1.  (2.8)

Proof. (2.4) is true since G_{p}(|T |x_{1}, …, |T |x_{p}) = G_{p}(T x_{1}, …, T x_{p}) for any (x_{1}, …, x_{p}) ∈ H^{p} .

(2.5) follows from Remark 2.1(a).

(2.6) follows from Remark 2.1(b) and (2.3).

(2.7): since ∀x_{1}, x_{2} ∈ H

G_{p}(T x_{1}, T x_{2}) = ∥T x_{1}∥^{2}∥T x_{2}∥^{2}−|(T x_{1}|T x_{2})|^{2} = ∥T x_{1}∥^{2}∥T x_{2}∥^{2}−|(T^{∗}T x_{1}|x_{2})|^{2},

and since by (2.1)

∫_{S} |(T^{∗}T x_{1}|x_{2})|^{2}ds(x_{2}) = \frac{1}{n}∥T^{∗}T x_{1}∥^{2}, ∀x_{1} ∈ H,

from (2.3) (using, of course, Fubini’s theorem) we get (2.7).

(2.8) follows from (2.3) and from the evident equality ∥(T^{∗}T − I)x∥^{2} = ∥T^{∗}T x∥^{2} − 2∥T x∥^{2} + ∥x∥^{2}, x ∈ H.

Remark 2.3. Fix T ∈ L(H, Y).

(1) By means of the direct integration of the Grammian it is possible to get a “coordinate free” expression for D_{p}(T) in terms of D_{1} similar to (2.7) also for p ≥ 3, e.g. we have

D_{3}(T) = D_{1}^{3}(T) − \frac{3}{n} D_{1}(T)D_{1}(T^{∗}T) + \frac{2}{n^{2}} D_{1}(T T^{∗}T).

However, the corresponding higher order combinatorial formula looks rather complicated. For this reason formula (2.12) from Proposition 2.4 is preferable.

(2) If n ≥ 2 and p = 2, then (2.7) shows that in (2.6) we have the equality if and only if T = 0 (compare with Remark 2.1(b)).

Proposition 2.4. Let dim H = n, p ≤ n, T ∈ L(H, Y) and (e_{1}, …, e_{n}) be any orthonormal basis of H. Then:

D_{1}(T) = \frac{1}{n}( ∑_{k=1}^{n} ∥T e_{k}∥^{2}) = \frac{1}{n}∥T ∥_{HS}^{2},  (2.9)

D_{p}(T) = \frac{1}{n^{p}} ∑_{j_{1},...,j_{p}=1}^{n} G_{p}(T e_{j_{1}}, … T e_{j_{p}}),  (2.10)

D_{p}(I) = \frac{n!}{(n − p)!n^{p}}  (2.11)

and

D_{p}(T) = \frac{p!}{n^{p}} ∑_{1≤j_{1}<···<j_{p}≤n} λ_{j_{1}}^{2} … λ_{j_{p}}^{2},  (2.12)

where λ_{1}, …, λ_{n} are the eigenvalues of the operator |T | := (T^{∗}T)^{\frac{1}{2}} listed according to their multiplicities.

Proof. Fix x ∈ H. We have x = ∑_{k=1}^{n}(x|e_{k})e_{k}. Then T x = ∑_{k=1}^{n}(x|e_{k})T e_{k} and

∥T x∥^{2} = ∑_{j,k=1}^{n}(x|e_{j})\overline{(x|e_{k})}(T e_{j}|T e_{k}).

Integrating this equality over S with respect to s and using (2.1) we get (2.9).

In the proof of (2.10) we will use the standard notation, namely, 𝔖_{p} will stand for the set of all permutations σ : {1, …, p} → {1, …, p}, and we will denote by sgn σ the sign of a fixed permutation σ. Fix finite sequences (a_{1}, …, b_{p}), (b_{1}, …, b_{p}) of the elements of H. We have the next “Parseval equality” for determinants, which follows from [1, p. V. 34, formula (26) and Prop. 5]:

det((a_{i}|b_{j})_{i,j=1}^{p}) = ∑_{1≤j_{1}<···<j_{p}≤n} (det((a_{i}|e_{j_{k}})_{i,k=1}^{p})) \overline{(det((b_{i}|e_{j_{k}})_{i,k=1}^{p}))}. (2.13)

Fix now a finite sequence (x_{1}, …, x_{p}) of the elements of H. Applying equality (2.13) to the matrix (T^{∗}T x_{i}|x_{j})_{i,j=1}^{p} we get

G(T x_{1}, …, T x_{p})

= ∑_{1≤j_{1}<···<j_{p}≤n} ∑_{σ,π∈𝔖_{p}} sgnσ · sgn π ∏_{k=1}^{p}(x_{k}|T^{∗}T e_{j_{σ(k)}})\overline{(x_{k}|e_{j_{π(k)}})}. (2.14)

Now integrating both sides of equality (2.14) with respect to the variables x_{1}, …, x_{p} and measure s × ⋯ × s and using formula (2.1) we obtain

D_{p}(T) = \frac{1}{n^{p}} ∑_{1≤j_{1}<···<j_{p}≤n} ∑_{σ,π∈𝔖_{p}} sgnσ · sgn π ∏_{k=1}^{p}(T e_{j_{π(k)}}|T e_{j_{σ(k)}})

= \frac{p!}{n^{p}} ∑_{1≤j_{1}<···<j_{p}≤n} G(T e_{j_{1}}, …, T e_{j_{p}}) = \frac{1}{n^{p}} ∑_{j_{1},...,j_{p}=1}^{n} G(T e_{j_{1}}, … T e_{j_{p}}). (2.15)

The second equality in (2.15) is true since it can be easily seen that the relation

∑_{σ,π∈𝔖_{p}} sgnσ · sgnπ ∏_{k=1}^{p}(y_{σ(k)}|y_{π(k)}) = p!G(y_{1}, …, y_{p})

holds for all finite sequences (y_{1}, …, y_{p}) of elements of Y.

Equality (2.11) follows from (2.10).

Equality (2.12) also follows from (2.10) since in (2.10) the elements e_{k}, k = 1, …, n can be taken the eigenvectors of T^{∗}T.

Remark 2.5. (1) The validity of the important equality (2.10) was pointed out in [6], while the proof is given in [5, p. 113]; the above-given proof is different and, in a sense, less “combinatorial”.

(2) Let

|||T |||_{p,e_{·}} := ( ∑_{j_{1},...,j_{p}=1}^{n} G_{p}(T e_{j_{1}}, …, T e_{j_{p}}))^{1/2}.

Then from equality (2.10), inequality (2.6) and equality (2.9) we get

|||T |||_{p,e_{·}}^{2} = n^{p}D_{p}(T) ≤ ∥T ∥_{HS}^{2p}.

This relation implies, in particular, that the value of |||T |||_{p,e_{·}} does not depend on a particular choice of an orthonormal basis e_{·} := (e_{1}, …, e_{n}) of H (compare with the next item).

(3) Suppose for a moment that H is an infinite-dimensional separable Hilbert space, T : H → Y is a continuous linear operator, (e_{n})_{n∈ℕ} is an orthonormal basis of H and put again

|||T |||_{p,e_{·}} : = ( ∑_{j_{1},...,j_{p}=1}^{∞} G_{p}(T e_{j_{1}}, …, T e_{j_{p}}))^{1/2}

= lim_{n→∞} ( ∑_{j_{1},...,j_{p}=1}^{n} G_{p}(T e_{j_{1}}, … T e_{j_{p}}))^{1/2}.

(3a) [3, p. 44, Prop. III.2.2] If T is a Hilbert–Schmidt operator, then

|||T |||_{p,e_{·}} ≤ ∥T ∥_{HS}^{p}  (2.16)

and the value of |||T |||_{p,e_{·}} does not depend on a particular choice of an orthonormal basis e_{·} := (e_{n})_{n∈ℕ} of H (note that inequality (2.16) follows from Remark 2.1(b), however the second statement now needs a separate proof).

(3b) [3, p. 42, Prop. III.2.1] If for given T there are a natural number p and an orthonormal basis e_{·} := (e_{n})_{n∈ℕ} of H such that |||T |||_{p,e_{·}} < ∞, then T is a Hilbert–Schmidt operator.

Using the functional D_{p}, the definition of the p-th order mean dilatation number δ_{p}(T) of an operator T ∈ L(H, Y), given in the introduction, can be rewritten as

δ_{p}(T) = (\frac{D_{p}(T)}{D_{p}(I)})^{\frac{1}{2}} .  (2.17)

Evidently, when p = 1, (2.17) gives

δ_{1}(T) = D_{1}^{1/2}(T) = ( ∫_{S} ∥T x∥^{2}ds(x))^{\frac{1}{2}} .  (2.18)

The “normalized” version of Proposition 2.4 looks as follows:

Proposition 2.6. Let dim H = n, p ≤ n, T ∈ L(H, Y) and (e_{1}, …, e_{n}) be any orthonormal basis of H. Then:

δ_{1}(T) = \frac{1}{\sqrt{n}}∥T ∥_{HS} = \frac{1}{\sqrt{n}}( ∑_{k=1}^{n} ∥T e_{k}∥^{2})^{\frac{1}{2}} ,  (2.19)

δ_{p}(T) = \binom{n}{p}^{− \frac{1}{2}} ( ∑_{1≤j_{1}<···<j_{p}≤n} λ_{j_{1}}^{2} … λ_{j_{p}}^{2})^{\frac{1}{2}} ,  (2.20)

where λ_{1}, …, λ_{n} are the eigenvalues of the operator |T | := (T^{∗}T)^{\frac{1}{2}} listed according to their multiplicities;

δ_{n}(T) = det(|T |).  (2.21)

Proof. (2.19) follows from (2.9).

(2.20) follows from (2.12).

(2.21) follows from (2.20) by putting p = n.

Remark 2.7. (a) Equality (2.21) can be derived directly from Remark 2.1(f), its validity was already noted in [6]. It shows that the normalization through D_{p}(I) in the definition of the pMD-number is natural.

(b) Equality (2.19) implies that the functional T → δ_{1}(T) is a norm on L(H, Y) with the following property:

\frac{1}{\sqrt{n}}∥T ∥ ≤ δ_{1}(T) ≤ ∥T ∥, ∀ T ∈ L(H, Y).  (2.22)

(c) Suppose p > 1, then the functional T → δ_{p}(T) is absolutely p-homoge-neous on L(H, Y) (this follows, e.g., from (2.20)). Also (2.5) implies that δ_{p}(·) vanishes on the operators with a rank < p.

(d) If for a given operator T ∈ L(H, Y) we have ∥T ∥ ≤ 1 and δ_{1}(T) = 1, then T is an isometry (this is easy to see).

The following assertion shows that isometric operators can be characterized only in terms of δ_{1} and δ_{2}.

Proposition 2.8. Let H be a finite-dimensional Hilbert space with dim H = n ≥ 2, Y be any Hilbert space and T : H → Y be a linear operator. The following statements are equivalent:

(i) T is an isometry.

(ii) δ_{p}(T) = 1 ∀p ≤ n.

(iii) δ_{1}(T) = δ_{2}(T) = 1.

Proof. The implications (i) ⇒ (ii) ⇒ (iii) are evident.

First proof of (iii) ⇒ (i). It is sufficient to verify that T^{∗}T = I. For this it is enough to show that δ_{1}(T^{∗}T − I) = 0 (since δ_{1} is a norm). Equalities (2.7) and (2.8) imply

δ_{2}^{2}(T) = \frac{n}{n − 1}δ_{1}^{4}(T) − \frac{1}{n − 1}δ_{1}^{2}(T^{∗}T)  (2.23)

and

δ_{1}^{2}(T^{∗}T − I) = δ_{1}^{2}(T^{∗}T) − 2δ_{1}^{2}(T) + 1.  (2.24)

Now it is clear that (iii) and (2.23) imply δ_{1}(T^{∗}T) = 1. Thus the equality δ_{1}(T) = 1 and (2.24) imply δ_{1}(T^{∗}T − I) = 0. Consequently, T^{∗}T = I.

Second proof of (iii) ⇒ (i).^{2} Let λ_{1}, …, λ_{n} be the eigenvalues of the operator |T | := (T^{∗}T)^{\frac{1}{2}} listed according to their multiplicities. Using (iii) and (2.20) we get

∑_{k=1}^{n} λ_{k}^{2} = nδ_{1}^{2}(T) = n, ∑_{1≤j_{1}<j_{2}≤n} λ_{j_{1}}^{2}λ_{j_{2}}^{2} = \binom{n}{2}δ_{2}^{2}(T) = \binom{n}{2}.

Then

∑_{k=1}^{n} λ_{k}^{4} = ( ∑_{k=1}^{n} λ_{k}^{2})^{2} − 2 ∑_{1≤j_{1}<j_{2}≤n} λ_{j_{1}}^{2}λ_{j_{2}}^{2} = n^{2} − 2\binom{n}{2} = n

and

∑_{k=1}^{n}(λ_{k}^{2} − 1)^{2} = ∑_{k=1}^{n} λ_{k}^{4} − 2 ∑_{k=1}^{n} λ_{k}^{2} + n = 0.

Consequently, λ_{k} = 1, k = 1, …, n and |T | = I.

Remark 2.9. It is interesting to note that in [6] only the validity of the implication (ii) ⇒ (i) was conjectured. We see that even (iii) implies (i).

The following assertion will be used in the next section.

Lemma 2.10. Let dim H = n, T : H → Y be a linear operator and p ∈ {1, …, n}. Put

c_{n,p} := D_{p}(I) = \frac{n!}{(n − p)!n^{p}}

and

r_{n}(p, T) := D_{1}^{p}(T) − D_{p}(T).

Then:

\frac{1}{p^{p−1}} ≤ c_{n,p} ≤ 1,  (2.25)
δ_{p}(T) ≤ p^{\frac{p−1}{2}} δ_{1}^{p}(T),  (2.26)
0 ≤ r_{n}(p, T) ≤ \frac{p!}{n}∥T ∥^{2p},  (2.27)
δ_{p}^{2}(T) = \frac{1}{c_{n,p}} (δ_{1}^{2p}(T) − r_{n}(p, T)).  (2.28)

2Suggested by the Referee.

Proof. Relation (2.25) is easy to verify. Inequality (2.26) follows from (2.6) and (2.25).

(2.27): Let λ_{1}, …, λ_{n} be the eigenvalues of the operator |T | := (T^{∗}T)^{\frac{1}{2}} listed according to their multiplicities. Using (2.9), Newton’s (polynomial) formula and (2.10) we can write:

(D_{1}(T))^{p} − D_{p}(T) = \frac{1}{n^{p}} (∑_{k=1}^{n} λ_{k}^{2})^{p} − \frac{p!}{n^{p}} ∑_{1≤j_{1}<···<j_{p}≤n} λ_{j_{1}}^{2} … λ_{j_{p}}^{2}

= \frac{p!}{n^{p}} ∑_{k_{1}+···+k_{n}=p} \frac{1}{k_{1}! … k_{n}!} λ_{1}^{2k_{1}} … λ_{n}^{2k_{n}}

− \frac{p!}{n^{p}} ∑_{k_{1}+···+k_{n}=p, max k_{i}=1} \frac{1}{k_{1}! … k_{n}!} λ_{1}^{2k_{1}} … λ_{n}^{2k_{n}}

= \frac{p!}{n^{p}} ∑_{k_{1}+···+k_{n}=p, max k_{i}>1} \frac{1}{k_{1}! … k_{n}!} λ_{1}^{2k_{1}} … λ_{n}^{2k_{n}}.

Consequently,

(D_{1}(T))^{p} − D_{p}(T) = \frac{p!}{n^{p}} ∑_{k_{1}+···+k_{n}=p, max k_{i}>1} \frac{1}{k_{1}! … k_{n}!} λ_{1}^{2k_{1}} … λ_{n}^{2k_{n}}.

From this equality, as λ_{k} ≤ ∥T ∥, k = 1, …, n, and

∑_{k_{1}+···+k_{n}=p, max k_{i}>1} \frac{1}{k_{1}! … k_{n}!} ≤ n^{p−1},

we obtain (2.27).

(2.28) follows from (2.27).

Remark 2.11. (1) By (2.20 ) it can be seen that for a given operator T the equality δ_{p}(T) = δ_{1}^{p}(T) holds if and only if T is a scalar multiplier of an isometry. In this connection the asymptotic versions behave better (see the next section).

(2) A simple relation between δ_{p}(T) and δ_{1}^{p}(T), can be written using the notation from the theory of outer products of Hilbert spaces. Namely, let H, Y be finite-dimensional Hilbert spaces, T ∈ L(H, Y) and ⋀^{p} T be the antisymmetric outer p-th power of the operator T. Then

δ_{p}(T) = \binom{n}{p}^{− \frac{1}{2}} ∥ ⋀^{p} T ∥_{HS} and δ_{p}(T) = δ_{1}(⋀^{p} T).  (2.29)

These formulas follow easily from the following relation, established in [5, p. 112– 113, the proof of Prop. VI.2.2]: ∥ ⋀^{p} T ∥_{HS}^{2} = \frac{n^{p}}{p!} D_{p}(T). In [5] it is also shown that the antisymmetric outer p-th power of a Hilbert–Schmidt (or a nuclear) operator T acting between infinite-dimensional Hilbert spaces is again a Hilbert– Schmidt (a nuclear) operator (cf. also [4, Th. 2]).

3. Infinite-Dimensional Case

In this section H will be an infinite-dimensional Hilbert space and Y will be another Hilbert space. As before, L(H, Y) will denote the set of all continuous linear operators T : H → Y.

Fix a natural number p and consider the collection M_{p} of all finite-dimensio-nal vector subspaces M ⊂ H with dim M ≥ p. This collection is a directed set by set-theoretic inclusion.

Fix an operator T ∈ L(H, Y). Then for any M ∈ M_{p} and the restriction T |_{M} of T to M, the pMD-number δ_{p}(T |_{M}) is defined. In this way T generates with the net (δ_{p}(T |_{M}))_{M∈M_{p}}. Let us denote the upper limit of this net by \overline{\overline{δ}}_{p}(T) and call it the asymptotic upper pMD-number of T. In a similar manner, let us denote the lower limit of this net by \underline{\underline{δ}}_{p}(T) and call it the asymptotic lower pMD-number of T. Therefore

\overline{\overline{δ}}_{p}(T) := lim sup_{M∈M_{p}} δ_{p}(T |_{M}) and \underline{\underline{δ}}_{p}(T) := lim inf_{M∈M_{p}} δ_{p}(T |_{M}).

By formula (2.26) of Lemma 2.10 we have

δ_{p}(T |_{M}) ≤ p^{\frac{p−1}{2}} ∥T ∥^{p} , ∀ M ∈ M_{p}.

This inequality shows that the net (δ_{p}(T |_{M}))_{M∈M_{p}} is bounded and, consequently, always

\underline{\underline{δ}}_{p}(T) ≤ \overline{\overline{δ}}_{p}(T) ≤ p^{\frac{p−1}{2}} ∥T ∥^{p} < ∞.

In general, the net (δ_{p}(T |_{M}))_{M∈M_{p}} may not be convergent (as we will see below). In the case of convergence we shall call the operator T asymptotically pMD-regular. For an asymptotically pMD-regular T we put

\overline{δ}_{p}(T) = lim_{M∈M_{p}} δ_{p}(T |_{M}),

and call \overline{δ}_{p}(T) the asymptotic pMD-number of T.

An asymptotically 1MD-regular operator T will be called asymptotically MDregular and its 1MD-number will be called asymptotic MD-number.

The next result is somewhat unexpected. Its validity was not predicted in [6].

Theorem 3.1. Let H be an infinite-dimensional Hilbert space, Y be a Hilbert space, T ∈ L(H, Y) and p > 1 be a natural number. Then

\overline{\overline{δ}}_{p}(T) = (\overline{\overline{δ}}_{1}(T))^{p} and \underline{\underline{δ}}_{p}(T) = (\underline{\underline{δ}}_{1}(T))^{p};  (3.1)

moreover, the operator T is asymptotically pMD-regular if and only if it is asymptotically MD-regular and in the case of asymptotic MD-regularity the equality

\overline{δ}_{p}(T) = (\overline{δ}_{1}(T))^{p}  (3.2)

holds.

Proof. Let M ∈ M_{p}. We can write using formula (2.28) of Lemma 2.10:

δ_{p}^{2}(T |_{M}) = \frac{1}{p!}\binom{dim M}{p}^{−1}(dim M)^{p} (δ_{1}^{2p}(T |_{M}) − r_{dim M}(p, T |_{M})) . (3.3)

Evidently,

lim_{M∈M_{p}} \frac{1}{p!}\binom{dim M}{p}^{−1}(dim M)^{p} = 1.  (3.4)

Since according to inequality (2.27) of Lemma 2.10

r_{dim M}(p, T |_{M})| ≤ \frac{p!}{\sqrt{dim M}} ∥T |_{M}∥^{2p} ≤ \frac{p!}{\sqrt{dim M}} ∥T ∥^{2p},

we also have

lim_{M∈M_{p}} r_{dim M}(p, T |_{M}) = 0.  (3.5)

From (3.3) via (3.4) and (3.5) the we get (3.1). The “moreover” part now is clear.

This theorem shows that in the case of an infinite-dimensional domain it is sufficient to study only the numbers \overline{\overline{δ}}_{1}(T), \underline{\underline{δ}}_{1}(T) and \overline{δ}_{1}(T).

To simplify the notation, for a given operator T ∈ L(H, Y) let us denote its

– asymptotic upper 1MD-number \overline{\overline{δ}}_{1}(T) by \overline{\overline{δ}}(T),

– asymptotic lower 1MD-number \underline{\underline{δ}}_{1}(T) by \underline{\underline{δ}}(T),

– asymptotic MD-number \overline{δ}_{1}(T) by \overline{δ}(T). Using formula (2.19) from Proposition 2.6 the definition of these numbers can

be formulated directly in terms of the Hilbert–Schmidt norm as follows:

\overline{\overline{δ}}(T) = lim sup_{M∈M_{1}} \frac{1}{\sqrt{dim M}} ∥T |_{M}∥_{HS}, \underline{\underline{δ}}(T) = lim inf_{M∈M_{1}} \frac{1}{\sqrt{dim M}} ∥T |_{M}∥_{HS}.

For an operator T ∈ L(H, Y) let us put

m(T) := inf{∥T x∥ : x ∈ H, ∥x∥ = 1}.

The number m(T) is sometimes called the lower bound of T.

The next statement implies, in particular, that a given operator T may not be asymptotically MD-regular.

Proposition 3.2. Let H, Y be infinite-dimensional Hilbert spaces and T : H → Y be a continuous linear operator. Then:

(a) For any infinite-dimensional closed vector subspace X ⊂ H the inequality

\overline{\overline{δ}}(T) ≥ m(T |_{X})

holds.

(b) If ker(T) is infinite-dimensional, then \underline{\underline{δ}}(T) = 0.

(c) If T is a partial isometry such that ker(T) and T (H) are both infinitedimensional, then \overline{\overline{δ}}(T) = 1, while \underline{\underline{δ}}(T) = 0 and so T is not asymptotically MD-regular.

Proof. (a) Fix an infinite-dimensional X ⊂ H, a finite-dimensional vector subspace M ⊂ H and put

β_{M} := sup{δ_{1}(T |_{N}) : N ∈ M_{1}, N ⊃ M}.

Let us show that

β_{M} ≥ m(T |_{X}).  (3.6)

To prove (3.6), fix a natural number n and an n-dimensional vector subspace X_{n} of X such that X_{n} ∩ M = {0} (such a choice is possible because M is finite-dimensional and X is infinite-dimensional). Let also M_{n} := M + X_{n} and let M′ be the vector subspace of M_{n} orthogonal to X_{n}. Using formula (2.19) from Proposition 2.6 we can write:

δ_{1}^{2}(T |_{M_{n}}) = \frac{∥T |_{M′}∥_{HS}^{2} + ∥T |_{X_{n}}∥_{HS}^{2}}{dim(M′) + n} ≥ \frac{∥T |_{X_{n}}∥_{HS}^{2}}{dim(M′) + n} ≥ \frac{n}{dim(M′) + n} m(T |_{X})^{2}.

As β_{M} ≥ δ_{1}(T |_{M_{n}}) and dim(M′) ≤ dim(M), we get

β_{M}^{2} ≥ \frac{n}{dim(M) + n} m^{2}(T |_{X}).  (3.7)

Since n is arbitrary, from (3.7) we have

β_{M}^{2} ≥ sup_{n} \frac{nm^{2}(T |_{X})}{dim(M) + n} ≥ lim_{n} \frac{nm^{2}(T |_{X})}{dim(M) + n} = m^{2}(T |_{X}).

This relation, together with (3.6) and the definition of \overline{\overline{δ}}(T), implies (a). (b) Let X = ker(T), fix a finite-dimensional vector subspace M ⊂ H and put

α_{M} := inf{δ_{1}(T |_{N}) : N ∈ M_{1}, N ⊃ M}.

Let us show that

α_{M} = 0.  (3.8)

To prove (3.8), fix a natural number n and an n-dimensional vector subspace X_{n} of X such that X_{n} ∩ M = {0} (such a choice is possible because M is finitedimensional and X is infinite-dimensional). Let also M_{n} := M + X_{n} and M′ be the subspace of M_{n} orthogonal to X_{n}. Using formula (2.19) from Proposition 2.6 and taking into account that T |_{X_{n}} = 0 we can write:

δ_{1}^{2}(T_{M_{n}}) = \frac{∥T |_{M′}∥_{HS}^{2} + ∥T |_{X_{n}}∥_{HS}^{2}}{dim(M′) + n} = \frac{∥T |_{M′}∥_{HS}^{2}}{dim(M′) + n} .

From this we get

α_{M}^{2} ≤ \frac{∥T |_{M′}∥_{HS}^{2}}{dim(M′) + n} .

Observe now that since dim(M′) ≤ dim(M) and ∥T |_{M′}∥_{HS} ≤ ∥T ∥\sqrt{dim(M′)}, we have

α_{M}^{2} ≤ lim_{n} \frac{∥T |_{M′}∥^{2}}{dim(M′) + n} = 0.

This relation together with the definition of \underline{\underline{δ}}(T) implies (b).

(c) follows from (a) and (b).

The last proposition motivates the following

Problem. Give a characterization of asymptotically MD-regular operators in terms of some other known parameters (in terms of the spectrum, in terms of the diagonal (for the diagonal operators), etc.).

Acknowledgement

We are grateful to the referee for helpful comments and suggestions. The third author acknowledges support from Xunta de Galicia (D.O.G. do

25-02-00) during his visit to Vigo.

References

1. N. Bourbaki, Espaces vectoriels topologiques. Masson, Paris, 1981.

2. A. Castejón, E. Corbacho, and V. Tarieladze, AMD numbers, compactness, strict singularity and essential spectrum of operators. Submitted to Georgian Math. J. in 2001.

3. E. Corbacho, Hipervolúmenes, su convergencia y sumabilidad en los operadores lineales acotados del espacio de Hilbert. Ph.D. thesis, Zaragoza, 1978.

4. S. Geiss, Antisymmetric tensor products of absolutely p-summing operators. J. Approx. Theory 68(1992), No. 3, 223–246.

5. A. Martínez, Normas tensoriales e ideales de operadores. Ph.D. thesis, Madrid, 1986.

6. J. L. Rubio de Francia, Unpublished notes.

7. N. Tomczak-Jaegermann, Banach–Mazur distance and finite dimensional operator ideals. Longman Scientific&Technical, Harlow, 1989.

(Received 15.01.2001; revised 15.06.2001)

Authors’ address:

Universidad de Vigo Departamento de Matemática Aplicada 36280 Vigo, Galicia, España E-mail: acaste@dma.uvigo.es

corbacho@uvigo.es visit01@dma.uvigo.es

The permanent address of V. Tarieladze: N. Muskhelishvili Institute of Computational Mathematics Georgian Academy of Sciences 8, Akuri St., Tbilisi 380093, Georgia E-mail: tar@gw.acnet.ge