Strictly singular operator

Strictly singular operators are dealt with in the mathematical sub-area of functional analysis . They are singular, that is, non- invertible , linear operators between Banach spaces with an additional, tightening property, which leads to the designation strictly singular .

definition

A continuous , linear operator between Banach spaces and is called strictly singular if there is no infinite-dimensional sub- Banach space , so that the restriction is a Banach space isomorphism, that is, a linear homeomorphism . ${\ displaystyle T: X \ rightarrow Y}$ ${\ displaystyle X}$ ${\ displaystyle Y}$ ${\ displaystyle Z \ subset X}$ ${\ displaystyle T | _ {Z}: Z \ rightarrow T (Z)}$

On finite-dimensional spaces, every operator is therefore strictly singular, because the defining condition is empty, since there are no infinite-dimensional subspaces at all. Even invertible operators are strictly singular here. This term only makes sense for infinitely dimensional spaces. Strictly singular operators are singular on such spaces, because with the names of the definition the condition must also apply to. But beyond that, the condition must apply to all infinite-dimensional sub-annex spaces, so that it is a question of a tightening of the singularity. ${\ displaystyle Z = X}$

Examples

Compact operators are strictly singular, because with the notation of the above definition every operator is also compact and therefore cannot be a Banach space isomorphism between infinite-dimensional Banach spaces.

{\ displaystyle T | _ {Z}: Z \ rightarrow T (Z)}

Weakly compact operators are generally not strictly singular; a simple counterexample is the identical operator on an infinitely dimensional reflexive space that is weakly compact and even invertible. But if it has the Dunford-Pettis property , then every weakly compact operator is strictly singular. ${\ displaystyle T: X \ rightarrow Y}$ ${\ displaystyle X}$ ${\ displaystyle T: X \ rightarrow Y}$

If , every continuous linear operator is strictly singular. In particular, the natural embedding of the sequence spaces is strictly singular but not compact. ${\ displaystyle 1 \ leq p <r}$ ${\ displaystyle T: \ ell ^ {p} \ rightarrow \ ell ^ {r}}$ ${\ displaystyle \ ell ^ {p} \ rightarrow \ ell ^ {r}}$

properties

Strictly singular operators are closely related to compact operators, the following theorem, which goes back to T. Kato , applies :

A continuous, linear operator between Banach spaces is strictly singular if and only if there is an infinite-dimensional sub- Banach space for every infinite-dimensional sub-Banach space , so that is compact. ${\ displaystyle T: X \ rightarrow Y}$ ${\ displaystyle X_ {1} \ subset X}$ ${\ displaystyle X_ {2} \ subset X_ {1}}$ ${\ displaystyle T | _ {X_ {2}}: X_ {2} \ rightarrow Y}$

Furthermore, strictly singular operators have the expected so-called ideal properties, that is:

The set of all strictly singular operators between two Banach is in the operator norm is a closed space of the area of continuous, linear operators and is a product of operators, one of which is strictly unique, so is the product strictly singular. ${\ displaystyle TS}$

Note, however, that the adjoint operator to a strictly singular operator is generally not strictly singular, here strictly singular operators do not behave like compact operators (see Schauder's theorem ). To construct a counterexample, note that every separable Banach space has the quotient of . So there is a quotient mapping which, according to the above examples, is strictly singular. The adjoint mapping is not strictly singular. ${\ displaystyle \ ell ^ {1}}$ ${\ displaystyle \ ell ^ {1} \ rightarrow \ ell ^ {2}}$ ${\ displaystyle \ ell ^ {2} \ rightarrow \ ell ^ {\ infty}}$

Individual evidence

↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , Definition 2.1.8
↑ YA Abramovich, CD Aliprantis: An Invitation to Operator Theory , Oxford University Press (2002), ISBN 0-821-82146-6 , definition 4.56
↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , Theorem 5.5.1
↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , set 2.1.9
↑ YA Abramovich, CD Aliprantis: An Invitation to Operator Theory , Oxford University Press (2002), ISBN 0-821-82146-6 , sentence 4.61
↑ YA Abramovich, CD Aliprantis: An Invitation to Operator Theory , Oxford University Press (2002), ISBN 0-821-82146-6 , Corollary 4.62
↑ YA Abramovich, CD Aliprantis: An Invitation to Operator Theory , Oxford University Press (2002), ISBN 0-821-82146-6 , Chapter 4.5, Task 6

[1] F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , Definition 2.1.8

[2] YA Abramovich, CD Aliprantis: An Invitation to Operator Theory , Oxford University Press (2002), ISBN 0-821-82146-6 , definition 4.56

[3] F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , Theorem 5.5.1

[4] F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , set 2.1.9

[5] YA Abramovich, CD Aliprantis: An Invitation to Operator Theory , Oxford University Press (2002), ISBN 0-821-82146-6 , sentence 4.61

[6] YA Abramovich, CD Aliprantis: An Invitation to Operator Theory , Oxford University Press (2002), ISBN 0-821-82146-6 , Corollary 4.62

[7] YA Abramovich, CD Aliprantis: An Invitation to Operator Theory , Oxford University Press (2002), ISBN 0-821-82146-6 , Chapter 4.5, Task 6