Vol. 92, No. 1, 1981

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Unitary equivalence to integral operators

Viakalathur Shankar Sunder

Vol. 92 (1981), No. 1, 211–215
Abstract

A bounded operator A on L2(X) is called an integral operator if there exists a measurable function k on X ×X such that, for each f in L2(X),

∫
|k(x,y)f(y)|dμ(y) < ∞ a.e.

and

        ∫
Af (x) =   k(x,y)f(y)dμ(y) a.e.

(Throughout this paper, (X,μ) will denote a separable, σ-finite measure space which is not purely atomic.) An integral operator is called a Carleman operator if the inducing kernel k satisfies the stronger requirement:

∫
|k(x,y)|2dμ (y) < ∞ for almost every x in X.

Mathematical Subject Classification 2000
Primary: 47B38
Secondary: 47G05
Milestones
Received: 5 January 1978
Published: 1 January 1981
Authors
Viakalathur Shankar Sunder