properties of the transpose operator

In this entry, whenever V,W are normed vector spacesPlanetmathPlanetmath, (V,W) denotes the algebra of bounded linear operators VW.

Let X,Y,Z be normed vector spaces and X,Y,Z be their continuous dual spaces. Let T,S(X,Y), R(Y,Z) and λ.

0.0.1 Basic properties

  • T(Y,X) and T=T.

  • (λT)=λT.

  • (S+T)=S+T.

  • (RT)=TR.

  • If T-1 exists and T-1(Y,X) then (T)-1(X,Y) and (T)-1=(T-1).

0.0.2 Miscellaneous properties

  • If we endow X and Y with the weak-* topologyMathworldPlanetmath then T:YX is continuous.

  • T is an isometric isomorphism if and only if T is an isometric isomorphism.

  • If T is compactPlanetmathPlanetmath ( then T is also compact.

  • If T is compact and Y is a Banach spaceMathworldPlanetmath, then T is also compact.

Title properties of the transpose operator
Canonical name PropertiesOfTheTransposeOperator
Date of creation 2013-03-22 17:36:02
Last modified on 2013-03-22 17:36:02
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 7
Author asteroid (17536)
Entry type Theorem
Classification msc 46-00
Classification msc 47A05
Synonym transpose operator properties