A note on some operator theory in certain semi-inner-product spaces.

Abstract

Let X be a smooth uniformly convex Banach space and let [ ⋅ , ⋅ ] [\cdot ,\cdot ] be the unique semi-inner-product generating the norm of X. If A is a bounded linear operator on X, A † {A^\dagger } mapping X to X is called the generalized adjoint of A if and only if [ A ( x ) , y ] = [ x , A † ( y ) ] [A(x),y] = [x,{A^\dagger }(y)] for all x and y in X. In this setting adjoint abelian (iso abelian) operators [5] are characterized as those operators A for which A † = A ( A † = A − 1 {A^\dagger } = A({A^\dagger } = {A^{ - 1}} , i.e. the invertible isometries). It is also shown that the compression spectrum of an operator is contained in its numerical range.