Schur Products of Operators and the Essential Numerical Range

Quentin F. Stout
EECS Department, University of Michigan

Abstract: Given an orthonormal basis E for a separable infinite-dimensional Hilbert space H, the Schur product of two bounded linear operators A and B on H with respect to E is the operator whose matrix entries are obtained by taking the termwise product of the matrix entries for A and B. It can be shown that the Schur product is a bounded linear operator on H, and hence Schur multiplication defines a new, commutative Banach algebra B_E on the bounded linear operators over H.

For any operator T it is shown that the following three conditions (and others) are equivalent:

0 is in the essential numerical range of T.
There is a basis E such that T is in the kernel(hull(compact operators)) in B_E.
There is a basis such that Schur multiplication by T is a compact operator in B(B(H)) mapping Schatten classes into smaller Schatten classes.

Thus we provide new results linking B(H), B_E, and B(B(H)).

Keywords: Schur multiplication, Hadamard multiplication, essential numerical range, matrix representations, Schatten classes, Hilbert space, trace, majorization, operator theory, analysis, mathematics

Complete paper. This paper appears in Transactions American Mathematical Society 264 (1981), pp. 39-47.

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