Full operator

Completely continuous operators are examined in the mathematical sub-area of functional analysis. There are certain linear operators between Banach spaces , which are closely related to the compact operators , they are also called Dunford-Pettis operators .

definition

A linear operator between two Banach spaces is called complete if the image of each set compact in the weak topology is compact in the norm topology of the image space.

In formulas: The linear operator between Banach spaces is called complete if the image is norm -compact for all weakly compact ones. ${\displaystyle T:X\rightarrow Y}$ ${\displaystyle K\subset X}$ ${\displaystyle T(K)}$

This definition goes back to Hilbert . More precisely, Hilbert used the following equivalent characterization:

A linear operator between Banach spaces is complete if and only if every weakly-convergent sequence is mapped onto a norm -convergent sequence .

Remarks

• Not all authors differentiate between consistency and compactness. Heuser uses these terms synonymously and means compact operators. This is particularly not uncommon if authors are only interested in operators on Hilbert spaces anyway . It is called C * -algebras , all of whose irreducible Hilbert space representations have their pictures in the compact operators, CCR-algebras , where CCR for completely Call continuous representation is. It is therefore recommended to check the definition of completeness used by the respective author.

• The space of complete operators between two Banach spaces is a closed subspace of the space of all bounded operators between these Banach spaces. Furthermore, the complete operators fulfill the ideal property, that is, a product of two operators between Banach spaces is complete as soon as one of the factors is.

• The term complete operator is used in the definition of the Dunford-Pettis property . This is why complete operators are also called Dunford-Pettis operators .

• Every continuous linear operator in a Banach space is complete, because according to a theorem of Issai Schur every weakly convergent sequence in is already norm -convergent . ${\displaystyle T:\ell ^{1}\rightarrow Y}$ ${\displaystyle Y}$ ${\displaystyle \ell ^{1}}$

Comparison with compact operators

→ Main article : Compact operator

Compact operators are defined in a very similar way; they require that the image of each bounded set is relatively compact , that is, has a compact closure .

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

${\displaystyle T{\text{ kompakt }}\Rightarrow T{\text{ vollstetig }}\Rightarrow T{\text{ stetig }}}$

The inversions do not apply. So the identity on the sequence space is complete but not compact. The identity on is continuous but not complete. ${\displaystyle \ell ^{1}}$ ${\displaystyle \ell ^{2}}$

In the above definition of continuity, norm compactness is required and not just relative compactness as in the definition of the compact operator. This makes no difference here, because if a complete operator is, then it is continuous with regard to the standard topologies (even if only relative compactness is required in the definition) and therefore also continuous with regard to the weak topologies. So the image of a weakly compact set is always weakly compact and therefore weakly closed, even more so standardized, so that relative compactness here already means standard compactness. ${\displaystyle T(K)}$ ${\displaystyle T}$ ${\displaystyle T(K)}$

For operators on reflexive Banach spaces with values ​​in any Banach spaces, the terms completeness and compactness coincide, because bounded sets and relatively weakly compact sets are identical in reflexive spaces.

Comparison with weakly compact operators

→ Main article : Weakly compact operator

The classes of the complete and the weakly compact operators both include the class of the compact operators and are both in the class of the continuous operators. However, there is no general relationship between complete and weakly compact operators.

According to the above, the identity on is complete, but it is not weakly compact, because otherwise the bounded unit sphere would be weakly compact and would be reflexive , which is not the case. ${\displaystyle \ell ^{1}}$ ${\displaystyle \ell ^{1}}$ ${\displaystyle \ell ^{1}}$

The identity on is weakly compact, because it is reflexive, but it is not complete, because otherwise the weakly compact unit sphere would be norm-compact and would be finite-dimensional, which is not the case. ${\displaystyle \ell ^{2}}$ ${\displaystyle \ell ^{2}}$ ${\displaystyle \ell ^{2}}$

There is an important class of spaces on which all weakly compact operators are complete, that is, the spaces with the Dunford-Pettis property .

Individual evidence

1. ↑ David Hilbert: Fundamentals of a general theory of the linear integral equations IV. In: Messages of the Royal Society of Sciences Göttingen , Mathematics-Physics. 1906, pp. 157-227.
2. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , sentence 3.4.36.
3. ↑ Harro Heuser: functional analysis. Teubner-Verlag 1975, ISBN 3-519-02206-0 , p. 76.
4. ^ W. Arveson: Invitation to C * -algebras. Springer, 1998, ISBN 0-387-90176-0 , definition 1.5.1.
5. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , exercise 3.49.
6. ↑ Terry J. Morrison: Functional Analysis: An Introduction to Banach Space Theory. Wiley 2001, ISBN 0-471-37214-5 , definition 6.6.
7. ↑ J. Schur: About linear transformations in the theory of infinite series. J. Reine Angewandte Mathematik 151 (1920), pp. 79-111.
8. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , p. 219 below.
9. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , sentence 3.4.37.
10. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , before definition 3.5.15.