A spectral mapping theorem for tensor products of unbounded operators

Abstract

Introduction.In this note we will discuss the spectrum of tensor products of not necessarily bounded operators on Banach spaces X and Y X ® Y will denote the tensor product of X and Y in some uniform cross-norm [1], Thus, (i) X ® y is the completion of the algebraic tensor product in a norm with ||x ® y\\ = ||x|| \\y\\; (ii) for any Ae5£{X\ the bounded operators on X, and Be^(Y\ there is an operator A® Be Se{X ® Y) with (A ®B)(x®y) = Ax® By and \\A ® B\\ = \\A\\ \\B\\.Typical examples of such uniform cross-norms are the usual Hilbert space tensor product norm and the LP norm on LGiven a polynomial (or a rational function) in two variables and closed operators A on X and B on Y, we want to discuss the spectrum of P(A ® ƒ, / ® B) as an operator on X ® Y For unbounded operators, one must define what it means for an operator C on X ® Y "to be" P(A ® 1,1 ® B).We take a fairly strong definition:DEFINITION 1.Given a closed operator A with nonempty resolvent set on a Banach space, X, we say that a sequence A n of bounded operators on X is an âiï{A)~approximation if and only if A n converges to A in norm resolvent sense [2] and each A n is a polynomial in resolvents of A.DEFINITION 2. Given closed operators A and B on Banach spaces X and X and a rational function, P(z, co\ we say that a closed operator C on X ®Y equals P(A ®I 9 I®B) (or P(A, B\ for short) if and only if, there exists an ^(A)-approximation, A m and an ^(B)-approximation, B n9 so that P(A m B n ) converges in norm resolvent sense to C.Existence and uniqueness questions for P(A 9 B) naturally arise.In applying Theorem 1 below, all the hard analysis is in proving that existence holds.The existence and uniqueness question is discussed in detail in a forthcoming paper [3], primarily in the case where A and B are generators of bounded holomorphic semigroups.In the general case, we do not know whether it is possible for two different operators C and C to both "equal" P(A 9 B) but in that case our proof of Theorem 1 implies that (C -k)~l -(C -A)" 1 is quasinilpotent.

Locations

• Project Euclid (Cornell University) - View - PDF
• Bulletin of the American Mathematical Society - View - PDF