An isomorphic characterization of the Schmidt class

C ^OMPOSITIO M ^ATHEMATICA

D. R. L ^EWIS

An isomorphic characterization of the Schmidt class

Compositio Mathematica, tome 30, n

^o

3 (1975), p. 293-297

<http://www.numdam.org/item?id=CM_1975__30_3_293_0>

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293

AN ISOMORPHIC CHARACTERIZATION OF THE SCHMIDT CLASS

D. R. Lewis 1

Noordhoff International Publishing

Printed in the Netherlands

This note

gives

^an

isomorphic

characterization of the Hilbert-Schmidt

norm within the class of

unitarily

^invariant crossnorms on the _{space R} of finite rank operators ^on

l2 . Specifically

^{if the}

completion

of R under a

has an unconditional basis

(or

^more

generally

local unconditional struc-

ture)

^{than a is}

equivalent

^{to the} Hilbert-Schmidt norm.

For convenience

only

^real Banach spaces ^are considered. Notation and

terminology

^is standard, ^{with the}

following possible exceptions.

For E finite dimensional the unconditional basis constant of E is denoted

by x(E),

^{and the} parameter q ^{is defined}

by ri(E)

⁼

inf ~u~~v~~(F),

^{where the}

infimum is taken over all _{spaces F} and

pairs of operators

^{u : E ~} F, ^{v : F - E}

satisfying

^{vu =}

identity.

The definition

of îl

is extended to infinite dimen- sional _spaces

by setting ~(G)

⁼

infDsuPE~D~(E),

with the infimum taken

over all confinal collections of finite dimensional

subspaces

^{of G. G has}

local unconditional structure if

~(G)

^~.

R

(respectively, R(n))

^{is the} space of finite rank operators

on l2 (resp., l2).

^{A norm a on R}

(or R(n))

^{is called a}

unitarily

^{invariant crossnorm}

if,

^for

each

u ~ R, ~u~ ~ 03B1(u) ~ i1(u)

^and

a(guh)

⁼

a(u)

for all isometries _g and h.

This coincides with Schatten’s definition

[7].

^The

completion

^{of R under}

a is written

R(a),

^and

R(a, n)

^is

R(n)

^{with norm a.}

Finally,

03C0p and

ip

^denote

the

p-absolutely summing

^and

p-integral

norms,

respectively ([6]).

THEOREM :

If a is

^a

unitarily

^invariant crossnorm on

12

^and

if R(a)

^{has local}

unconditional structure, then ^{a is}

equivalent

^to the Hilbert-Schmidt norm.

PROOF : Fix n and let P

(respectively, Q)

be the natural inclusion

of R(a, n) (resp., R(a, n)’)

^into

R(n2’ n).

^G denotes the _group of isometries

of l2 and dg

is the normalized Haar measure on G. Let Q) E

R(n)

^with

03C9 ~

0, ^{and write}

1 The author was supported ⁱⁿ part by NSF GP-42666.

294

(03C9*03C9)1 2 =

^where

(ei)

^{is some} orthonormal basis

for l2

^and

each

Ai

~ ^0. The dual of

R(a, n)

^is

naturally

identified with

R(a’, n),

^where

a’ is the associate crossnorm of

a([7])

^{and the} action is the trace of the com-

position.

Define a

probability

measure y ^on the closed unit ball

of R(a, n) by setting

Then for u E

R(a, n)

the first since Q) _may be translated to

(03C9*03C9)1 2 by

^an

isometry.

^Now in the last

integral replace g by

gg£, where 8 ⁼

(03B51, ···, 8n)

^{is an}

n-tuple

^of

signs

^and

g03B5(ei)

= 03B5iei.

Averaging

^{over all such}

n-tuples 8

^and

applying

^Khinchin’s

inequality yields

for _any vector

e ~ ln2

^{with norm one. Let dm} be the normalized

(n -1 )-di-

mensional rotational invariant measure on the unit

sphere of 1’2

^and en the constant

satisfying

This

equality

^shows

that for ^~e~

^{= 1}

and the last is

nothing

^but cn

21C2(U). Combining inequalities

^now

gives

so that

To

estimate 7rl(P)

^let y be the

probability

measure on the closed unit ball of

R(a, n)’

^defined

by

where e is some fixed vector of norm one. The

previous inequality

^shows

for

^all

MëR(n),

^and

consequently 03C01(P) ~ cn . By

^the

proof of Theorem

^3.5

of [2]

and by

^{, so}

This first

inequality

^{is also valid if a is}

replaced by

its associate a’, ^and hence

by duality (using 03C0’2

⁼

712)

The constants in the last two

inequalities

^{are both dominated}

by 03C02~(R(03B1)),

so that ^a and _03C02 are

03C04~(R(03B1))2 - equivalent

^{on R. This}

completes

^the

proof.

296

REMARK 1: An

absolutely summing

operator ^{on a}

subspace

^{of an}

L1(03BC)-

space ^{or a}

quotient

^{of a}

C(K)-space

^{must factor}

through

^an

L1-space [2].

Thus the same

proof

^as above shows that a is

equivalent

^to the Hilbert- Schmidt norm if

R(a)

^is

isomorphic

^{to a}

subspace

^{of an}

L1-space

^{or to a}

quotient

^{of a}

C(K)-space.

^In

addition, by applying

^a basis selection theorem of

Kadec-Pelczynski [3]

^{and the result of}

McCarthy [5]

^that

R(cp) Í Lp,

it can be shown that ods

equivalent

^{to n2}

if R(a)

^c=

L p for some p

&#x3E; 2. It seems

a reasonable

conjecture

^{that a is}

equivalent

to 1r2 whenever

R(a)

^embeds

in a space with an unconditional basis.

REMARK 2: Given 03BB ~ ^{1 there is a}

unitarily

^invariant crossnorm a on R which is

equivalent

^{to 1r2} and such that the unconditional basis constant

of R(a)

^{is at} least 03BB. To see this fix n and let

~n

^{be the}

symmetric

gauge function

([7], Chapter V)

^on the space

of finitely

^{non-zero scalar} sequences defined

by

The crossnorm a on R induced

by 0,,

^is

clearly n2-equivalent

^{to 03C02 and}

by

the

inequalities

^{of the}

proof ~(R(03B1)) ~

03C0-2n1 2.

Similarly

^{it is}

possible

^to

produce unitary crossnorms 03B2

^{for which}

~(R(03B2, n))

^increases very

slowly.

For

instance,

^{the crossnorm}

03B2

^induced

by

^the

symmetric

gauge function

is, asymptotically, (log n) 2-equivalent

^{to 03C02 on}

ln2.

REMÂRK3:

Suppose R(a, n)

^{has a basis with the} property ^that each per- mutation of the basis vectors

naturally

^{defines an}

isomorphism

^{of norm at}

most 03BB.

By [4] q(R(a, n)) ~ 303BB,

^{so a is}

9n4 À2-equivalent

^{to 2 on}

ln2.

REFERENCES

[1] ^{Y. GORDON: On} p-absolutely summing ^{constants of Banach} spaces. Israeal J. Math. 7

(1969) ^151-162.

[2] ^{Y. GORDON and} D. R. LEWIS: Absolutely summing operators and local unconditional structures. Acta Math. (to appear).

[3] M. I. KADEC and A. OELCZYNSKI: Bases, lacunary sequences and complemented subspaces ^{in the} space Lp. Studia Math. 21 (1962) ^161-176.

[4] D. R. LEWIS: A relation between diagonal and unconditional basis constants (to appear

[5] ^Ch. A. MCCARTHY: Cp. ^{Israel J. Math. 5} (1967) ^249-271.

[6] A. PLETSCH: Theorie der operatorenideale. Jena, 1972.

[7] R. SCHATTEN: Norm ideals of completely continuous operators. Berlin-Göttingen- Heidelberg, ^1960.

(Oblatum 11-XI-1974) Department ^of Mathematics

University ^{of Florida}

Gainesville, Florida 32601 U.S.A.