Weakly compact operator

Weakly compact operators are examined in functional analysis. It is a class of linear restricted operators between Banach spaces with an additional compactness property that is modeled on the compact operators . This concept formation plays an important role in the Dunford-Pettis property .

definition

Be and Banach spaces. A linear operator is called weakly compact if, for every bounded set, the weak closure of the image is weakly compact. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle T\colon X\rightarrow Y}$ ${\displaystyle B\subset X}$ ${\displaystyle {\overline {T(B)}}^{w}}$

If you replace the weak topology with the standard topology in this definition, which goes back to S. Kakutani and K. Yosida , you get exactly the concept of the compact operator.

properties

For a linear operator between Banach spaces we have: ${\displaystyle T\colon X\rightarrow Y}$

${\displaystyle T}$compact operator weakly compact operator bounded operator. ${\displaystyle \Rightarrow \,T}$ ${\displaystyle \Rightarrow \,T}$

The inversions do not hold, as the identical operators on the sequence spaces and show. ${\displaystyle \ell ^{1}}$ ${\displaystyle \ell ^{2}}$

• ${\displaystyle {\rm {id}}_{\ell ^{1}}}$ is restricted, but not weakly compact.
• ${\displaystyle {\rm {id}}_{\ell ^{2}}}$ is weakly compact, but not compact.

If and are Banach spaces, at least one of which is reflexive , then every bounded linear operator between them is weakly compact. ${\displaystyle X}$ ${\displaystyle Y}$

Sums, scalar multiples and norm limit values ​​of weakly compact operators are again weakly compact. A product of bounded linear operators is weakly compact if one of the factors is weakly compact or . The set of all weakly compact operators between the Banach spaces and is therefore again a Banach space with respect to the operator norm . In the case there is a closed two-sided ideal in the Banach algebra of all bounded operators on . ${\displaystyle ST}$ ${\displaystyle S}$ ${\displaystyle T}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X=Y}$ ${\displaystyle X}$

Characterizations

The following simple theorem characterizes weak compactness:

For a linear operator between Banach spaces are ${\displaystyle T\colon X\rightarrow Y}$

the following statements are equivalent:

• ${\displaystyle T}$ is weakly compact.
• ${\displaystyle T(\{x\in X;\|x\|\leq 1\})}$is relatively weakly compact .
• Every bounded sequence in has a subsequence such that in weakly converges. ${\displaystyle (x_{n})_{n}}$ ${\displaystyle X}$ ${\displaystyle (x_{n_{m}})_{m}}$ ${\displaystyle (Tx_{n_{m}})_{m}}$ ${\displaystyle Y}$

In the following characterization, which goes back to VR Gantmacher (for the case of separable spaces) and Nakamura (for the general case), denotes the canonical embedding in the dual space . ${\displaystyle j_{X}}$ ${\displaystyle X''}$

For a linear operator between Banach spaces, the following statements are equivalent: ${\displaystyle T\colon X\rightarrow Y}$

• ${\displaystyle T}$ is weakly compact.
• ${\displaystyle T''(X'')\subset j_{Y}(Y)\subset Y''}$.

Gantmacher's theorem

In analogy to Schauder's theorem , the following applies

Gantmacher's theorem : For a linear operator between Banach spaces, the following statements are equivalent: ${\displaystyle T\colon X\rightarrow Y}$

• ${\displaystyle T}$ is weakly compact.
• The adjoint operator is weakly compact. ${\displaystyle T'\colon Y'\rightarrow X'}$

From this one can derive a further characterization: For a linear operator between Banach spaces, the following statements are equivalent: ${\displaystyle T\colon X\rightarrow Y}$

• ${\displaystyle T}$ is weakly compact.
• ${\displaystyle T'\colon Y'\rightarrow X'}$is weak * -weak-steady.

Factoring via reflexive spaces

One says that a continuous, linear operator factors over a Banach space if there are continuous linear operators and gives with . Since a continuous, linear operator between two Banach spaces, one of which is reflexive, is weakly compact according to the above properties and since products of continuous linear operators are weakly compact if at least one factor is weakly compact, every continuous , linear operator that factors over a reflexive space, be weakly compact. According to a theorem by Davis , Figiel , Johnson and Pełczyński , the converse is also true, that is, one has the following characterization of weakly compact operators: ${\displaystyle T\colon X\rightarrow Y}$ ${\displaystyle Z}$ ${\displaystyle S\colon X\rightarrow Z}$ ${\displaystyle R\colon Z\rightarrow Y}$ ${\displaystyle T=R\circ S}$

• A continuous, linear operator is exactly weakly compact if it factors over a reflexive Banach space. The norms of the factors can be limited by twice the norm of the output operator.

Weakly compact operators on C (K)

Let it be a compact Hausdorff space and let it be the function space of continuous functions with the supreme norm . Then the weakly compact operators with values ​​in a Banach space can be given as follows: ${\displaystyle K}$ ${\displaystyle C(K)}$ ${\displaystyle K\rightarrow \mathbb {R} }$ ${\displaystyle T\colon C(K)\rightarrow X}$ ${\displaystyle X}$

Let it be a regular, vectorial measure on (with Borel's σ-algebra ) with values ​​in . Regularity here means that the scalar measures are regular for all . Then it's through ${\displaystyle \mu }$ ${\displaystyle K}$ ${\displaystyle X}$ ${\displaystyle \varphi \circ \mu }$ ${\displaystyle \varphi \in X'}$

${\displaystyle T_{\mu }f:=\int _{K}f\mathrm {d} \mu }$

given a weakly compact operator . The operator norm of is equal to the semivariation of the measure . ${\displaystyle T_{\mu }\colon C(K)\rightarrow X}$ ${\displaystyle T_{\mu }}$ ${\displaystyle \mu }$

Conversely, every weakly compact operator has this form, that is, there is a regular vectorial measure on with values ​​in , so that the operator is described by the above formula, that is, it holds . ${\displaystyle T\colon C(K)\rightarrow X}$ ${\displaystyle \mu }$ ${\displaystyle K}$ ${\displaystyle X}$ ${\displaystyle T=T_{\mu }}$

Such a weakly compact operator is compact if and only if is relatively compact. With this one can easily construct further examples of weakly compact operators that are not compact. ${\displaystyle T_{\mu }\colon C(K)\rightarrow X}$ ${\displaystyle \{\mu (A)|A\subset K{\text{ Borel-messbar}}\}\subset X}$

Individual evidence

1. ^ Robert E. Megginson: An Introduction to Banach Space Theory , Springer-Verlag 1998, ISBN 0-387-98431-3 , definition 3.5.1
2. ^ Robert E. Megginson: An Introduction to Banach Space Theory , Springer-Verlag 1998, ISBN 0-387-98431-3 , Theorem 3.5.8
3. ^ Robert E. Megginson: An Introduction to Banach Space Theory , Springer-Verlag 1998, ISBN 0-387-98431-3 , Theorem 3.5.13
4. ^ WJ Davis, T. Figiel, WB Johnson, A. Pełczyński: Factoring weakly compact operators , J. Functional Analysis (1974), Volume 17 No. 3, pp. 311-327
5. ↑ P. Wojtaszczyk: Banach spaces for analysts , Cambridge University Press, 1991, ISBN 0-521-35618-0 , II.C.5
6. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , Theorem 5.25
7. ^ Raymond A. Ryan: Introduction to Tensor Products of Banach Spaces , Springer-Verlag 2002, ISBN 1-85233-437-1 , Theorem 5.27