User:Multiplerre/Collectively compact linear operators

ltogether COMPACT SETS OF LINEAR OPERATORS

PHILIP MARSHALL ANSELONE AND THEODORE WINDLE PALMER

Vol. 25, No. 3 November 1968

PACIFIC JOURNAL OF MATHEMATICS

Vol. 25, No. 3, 1968

Altogether COMPACT SETS OF LINEAR

Administrators

P. M. ANSELONE AND T. W. PALMER

A lot of straight administrators from one normed direct space

to another is altogether minimal if and just if the association of

the pictures of the unit ball has conservative conclusion. This paper

concerns general properties of such sets. A few valuable cri

teria for sets of direct administrators to be all in all minimized

are given. Specifically, every minimal arrangement of minimized straight

administrators is on the whole minimal. As a fractional talk, each

by and large smaller arrangement of self adjoint or ordinary administrators

on a Hubert space is completely limited.

Leave X and Y alone genuine or complex normed direct spaces and [X, Y]

the space of limited direct administrators on X into Y. It is accepted that

[X, Y] has the standard geography aside from in Proposition 2.1(c), where a

solid conclusion shows up.

Let £g mean the shut unit ball in X. At that point 3ίT c [X, Y] is

all in all minimal if and just if the set Sγ^ = {Kx: Ke J>γ,

x e ^) has conservative conclusion in Y. By and large minimized sets and their

applications to indispensable conditions have been treated in various

papers [1-5, 7-9, 11-12]. Results acquired in this paper are utilized in

a spin-off [6] which relates ghastly properties of administrators T and Γw

,

n = 1, 2, . , with the end goal that Tn

— T emphatically and {Tn

- T} is by and large

smaller.

As often as possible it will be important to show that a set in Y or [X, Y]

is smaller. For this reason, review that a subset of a measurement space

is minimized if and just in the event that it is shut and consecutively reduced if and

just on the off chance that it is finished and completely limited (for each ε > 0 it has a

limited ε-net). A frequently valuable truth is that a set is completely limited

at whatever point it has a completely limited ε-net for each ε > 0. The natural

suggestion that a consistent capacity from one topological space to

another guides minimized sets onto smaller sets will be utilized a few

times. The accompanying speculation of the Arzela-Ascoli hypothesis will

be required.

LEMMA 1.1. Leave g alone an equicontinuous set of capacities from

a smaller measurement space J3t~ into a measurement space. For every p e 3ίγ,

accept that the set %p — {f(p): f £ J%γ} has minimized conclusion. At that point

the set gJΓ" = {f(p): f eg, pe <3f] has minimized conclusion.

2. General properties of all in all minimized sets* Collectively

minimized arrangements of administrators have various properties closely resembling

417

418 P. M. ANSELONE AND T. W. PALMER

those of sets with reduced conclusion in self-assertive normed direct spaces.

For instance, any subset or scalar different of an all things considered reduced

set is all things considered minimal. Any limited association or total of all in all

conservative sets is all things considered minimal. An all in all conservative set is

fundamentally limited.

Suggestion 2.1. Let SΓa[X, Y] be on the whole minimal. At that point

the accompanying sets are all things considered reduced:

(a) The arched structure of Jf\

(b) The surrounded structure {XK: | λ | ^ 1, Ke ST} of

( c ) The solid conclusion ^%γ* and standard conclusion ^ P of

(d) {ΣίU KKn

Kn

e 3T, ΣSU \K\ ^ b) for every b > 0, N ^

Verification. Mazur's hypothesis [10, p. 416] yields (a). The orbited body

of a smaller set in Y is conservative since the guide/characterized by/(λ, y) = Xy

is persistent. This yields (b). Since ^T c JP S

what's more, Sϊ~s

έ3 c

(c) is legitimate. Since 3ίγ is limited, the set in (d) has a place with the standard

conclusion of the arched hovered structure of bS%~.

The following outcome includes integrals of administrator esteemed capacities.

Let Γ be a limited stretch if X is genuine and a rectifiable curve if X is

complex. Assume Ka

(X) c [X, Y] for Xeγ and an of every a list set A.

For each ae An accept that \ Ka

(X)dX is the solid or standard constraint of

the standard approximating entireties,

Recommendation 2.2. With the previous documentation, expect that {Ka

(x):

a G A, Xeγ} is all things considered minimal. At that point l\ Ka

(X)dX: ae A> is col

lectively minimal.

Verification. This follows from Proposition 2.1(c), (d) and

Σ I λi

- \ - I 1 ^ length (Γ) .

3 = 1

For the following suggestion, let Z be another normed straight space.

Suggestion 2.3. Let ST c [X, F], ^/S c [Z, X] and ^ r c [γ, Z].

At that point

(a) ^%^ all things considered minimal, Λ? limited =» *f%γ^/έ all things considered

minimal,

(b) J ^ all things considered minimal, J{r reduced =>yΓJΓ on the whole

minimal.

All things considered COMPACT SETS OF LINEAR OPERATORS 41S

Verification, (a) Suppose \\M\\ <r for all Me^/f. At that point

c 3tr&

so J>f ^/S& is minimal and 3ίγ^-/ίf is all in all reduced, (b) Define

a guide/: Jϊr x 3tTέ% - +Z by f(N, y) = Λfy for Ne^F', y e Jϊγέ§.

Since ^//' and 5iγ& are minimal and/is constant, its range, which

contains ^SίΓ^f, is smaller. Subsequently Λ*5ϊί& is minimized and Λ^SίΓ

is all things considered conservative.

An all things considered conservative set is a limited arrangement of minimized administrators.

The opposite bombs as can be seen by considering the arrangement of one di

mensional projections of standard one in any interminable dimensional Banach

space. Anyway we have:

Hypothesis 2.4. Each conservative set S>ίί of minimal administrators in

[X, Y] is on the whole smaller.

Confirmation. Characterize maps fx

ST-*Yby fx

(K) = Kx for K e _3γ, xe&

furthermore, let g - {/.: x e .^}. Since \\fx

(K, - K2

) \\ ^ || Kt

- K2

1|, g is equi

ceaseless. Since each KeJ%Γ is smaller, the sets $K = K& are

conservative. By speculation, Sίί is smaller. Consequently, by Lemma 1.1,

%<f%γ = S^f.^J is conservative and 3ίγ is all things considered minimal.

Hypothesis 2.5. /Y is finished, at that point each completely limited set

of smaller administrators in [X, Y] is by and large minimal.

Verification. For this situation, Jsf is a reduced arrangement of minimal administrators.

By Theorem 2.4, Sίί is all things considered reduced. Consequently, J ^ is aggregate

ly smaller.

The banters of Theorems 2.4 and 2.5 are bogus:

Model 2.6. Leave 3ίί alone the arrangement of administrators on lp

(l ^ p ^ oo)

characterized by Kn

x — xnφlf n}>l. Since 5ίγ& is limited and one

dimensional, J%γ is altogether smaller. Yet, S>f isn't completely bound

ed, for || Km

- Kn

\\ = 2ιl

» if m Φ n.

Fractional banters of Theorem 2.5 are given in the following area.

3. Administrators on a Hubert space. All through this segment, let

X be a Hubert space. It will be indicated that each all things considered com

agreement set of self adjoint or ordinary administrators in [X, X] is completely

limited.

We start by thinking about arrangements of projections. Let £f — {x: || x \\ — 1}

furthermore, for every x e £f, let Ex

be the self adjoint projection onto the

,subspace spread over by x.

420 P. M. ANSELONE AND T. W. PALMER

LEMMA 3.1. Let ^ c Sf and . /= {Ex

x e ^/}. (Therefore, ^ can

be any arrangement of self adjoint projections with one-dimensional extents.)

The accompanying explanations are proportional:

(a ) ^ is completely limited;

(b) ^/S is completely limited;

( c) ^/? is on the whole conservative.

Confirmation. Since Ex

y = (y, x)x for yeX and xeS^, the guide/:

£S-+[X, X] given by f(x) = Ex

is ceaseless. Since ^/S

(a) suggests (b). By Theorem 2.5, (b) suggests (c). Since ^ c

(c) suggests (a).

LEMMA 3.2. Le£ ^/Z be an on the whole conservative arrangement of self adjoint

projections and Λ?' any subset comprising of commonly symmetrical projections.

At that point ^r?f

is limited and there is a number n, autonomous

of ^fίf', with the end goal that

Σ diminish EX

Evidence. Since ^f £f is completely limited, it very well may be secured by a

limited number n of open bundles of span 1/2. In the event that x, y e ^//'S? what's more, x _L y

at that point \\x — y\\ = ~\/2,so that x and ^/lie in various balls. The lemma

follows.

LEMMA 3.3. Assume X is a genuine Hilbert space and X is its

complexification characterized in the typical manner. For J%γ cz[X, X], let

J%γ c [X, X] be the arrangement of sanctioned expansions of administrators in 3ίγ.

At that point 5$γ is all things considered minimized if and just if Jϊϊγ is all in all

conservative.

Since the confirmation is clear, it is overlooked.

We are presently prepared to set up the chief aftereffects of this segment.

Hypothesis 3.4. Let ^γ be a lot of self adjoint or ordinary reduced

administrators on a Hilbert space. At that point the accompanying articulations

are equal:

( a ) J%γ is aggregately minimal.

(b) JίT* = {K*: Ke^T) is aggregately minimal.

(c ) J%γ is completely limited.

Evidence. Without loss of sweeping statement, X is mind boggling. Expect

3ίγ = {Ka

aeA}

aggregately minimal. At that point each Ka

is minimized. For each aeA,

Aggregately COMPACT SETS OF LINEAR OPERATORS 421

the phantom hypothesis yields a deterioration

with the self adjoint projections Ean multually symmetrical and with

dimίϊ^X^ 1 (in this way, the Xan are not really unmistakable). Since

is limited, there exists 6 < °o to such an extent that

For a € An and ε > 0, let

Naε = {n: \\an\^e},

^ = {Ean: aeA,ne Nas} ,

Λ α e

= 2J ^ccnEan >

neNaε

j%T = {Kaε: aeA} .

At that point KaEan = XanEan and, for n e Naε, Ean& = \~l

nKaEan^ c e~l

Ka&>

In this way, ^C ^ c e~ι

J3γέ2? also, ^C is on the whole conservative. Bj

Lemma 3.1, ^/C is completely limited. By Lemma 3.2, there exists n

to such an extent that, for each ae A, Naε contains close to nε

components

At that point J3ft is in the curved hovered structure of b nε

^/Sε

, so <_%f is completely

limited. Since \\Ka

— Kaε\\ < ε for all aeA, j%7 is a ε-net foi

In this way, Jίγ is completely limited.

This outcome and Theorem 2.5 give:

on the whole minimal if and just if Jγ~ completely limited .

Since || T* || = || Γ|| for all Te [X, X],

3f completely limited if and just if J^ * completely limited .

The hypothesis follows.

Hypothesis 3.5. Let 3γ be a lot of smaller administrators on a Hilber

space. At that point J%γ is completely limited if and just if both Sγ and

are all things considered conservative.

Verification. As above, Jγ* completely limited suggests Sγ and J%γ* col

lectively smaller. Presently expect 3γ and 3f* all things considered minimal

At that point the sets

and - {K + if*: Ke Sγ} , J? = {K - K*: KeST}

are all things considered conservative. By Theorem 3.4, and ^ are totall;

limited. Since 3ίγ U ^γ* c ( ^ + J^) U ( ^ - ^) , both Sγ an<

are completely limited.

422 P. M. ANSELONE AND T. W. PALMER

For the all things considered conservative set J%γ in Example 2.6 with p = 2,

Secondary sources: