English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Let E = (en}00 I be an orthonormal basis for a Hilbert space 9C. For operators A and B having matrices (a,),X, and (b.)XO., their Schur product is defined to be (aijbU), _. This gives f (9C) a new Banach algebra structure, denoted '13&. For any operator T it is shown that T is in the kernel (hull(compact operators)) in some '13& iff 0 is in the essential numerical range of T. These conditions are also equivalent to the property that there is a basis such that Schur multiplication by T is a compact operator mapping Schatten classes into smaller Schatten classes. Thus we provide new results linking f (9C), '%& and 1%(f (9C)). Let 'C be a separable infinite-dimensional Hilbert space, and A, B be bounded operators on 'K. For an orthonormal basis & = {en: 1 < n < oo) of JC, A and B have matrix representations (aij),j=1 and (b),..= . In 1911 Schur [15, p. 8] proved that the termwise product (a.b&),j 1, denoted A * B, is itself a bounded operator with norm satisfying IA * B ISI < I A B . This multiplication will herein be called Schur multiplication, as opposed to the more common, but falsely crediting, "Hadamard multiplication". Schur's result shows that once a basis & is selected, then * defines a commutative Banach algebra, denoted by ?5g. While there is a surge of interest in applications of Schur multiplication (see, for example, Bennett [3], Johnson and Williams [6], Pommerenke [11], or Styan [18]), litle is known about the structure of "I;F In particular, few connections between the structure of B (S) and ,I; have been explored, though Varopoulos [20] has asked whether the linear structure of I (SC) forces ',; to be a Q-algebra. Except for Bennett [4] and Ruckle [13], the systematic study of Schur multiplication as a means of obtaining unusual operators in 1 ( 13(J)) is a relatively untouched concept, even though several interesting operators are Schur multipliers, e.g., Kwapien and Pelczynski [8]. It is important to keep in mind that A * B depends upon a choice of basis. The symbol As will be used to denote the element of I (@ (Xo)) defined by A (B) = A * B, where * is with respect to the basis 6. It will be shown that the following conditions (and several others) are equivalent for any operator T. (i) 0 is in the essential numerical range of T. (ii) There exists a basis & such that T is in kernel(hull(fC;)), where '3C; is the ideal of compact operators in 613,;. Received by the editors October 19, 1979 and, in revised form, March 3, 1980. 1980 Mathematics Subject Classification. Primary 47A12, 47D99; Secondary 46J99.

Schur products of operators and the essential numerical range