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, ${A^\dagger }$ mapping X to X is called the generalized adjoint of A if and only if $[A(x),y] = [x,{A^\dagger }(y)]$ for all x and y in X. In this setting adjoint abelian (iso abelian) operators [5] are characterized as those operators A for which ${A^\dagger } = A({A^\dagger } = {A^{ - 1}}$, i.e. the invertible isometries). It is also shown that the compression spectrum of an operator is contained in its numerical range.