Operator synthesis II: Individual synthesis and linear operator equations

Abstract

The second part of our work on operator synthesis deals with individual operator synthesis of elements in some tensor products, in particular in Varopoulos algebras, and its connection with linear operator equations. Using a developed technique of "approximate inverse intertwining" we obtain some generalizations of the Fuglede and the Fuglede-Weiss theorems and solve some problems posed in [O, W2, W3]. Additionally, we give some applications to spectral synthesis in Varopoulos algebras and to partial differential equations. IntroductionThis work is a sequel of [ShT] where the problems of operator synthesis were treated "globally" for lattices of subspaces, bilattices, or, in coordinate setting, for subsets of direct products of measure spaces. Here we consider operator-synthetic properties of elements of some tensor products, first of all of the Varopoulos algebras V (X, Y ) = C(X)⊗C(Y ). The topic is deeply connected to the theory of linear operator equations and, more generally, to the spectral theory of multiplication operators in the space of bounded operators and in symmetrically normed ideals of operators. We obtain some extensions of the Fuglede and Fuglede-Weiss theorems, answer several questions posed in [O, W2, W3], give applications to spectral synthesis in Varopoulos algebras and (somewhat unexpectedly) to partial differential equations.Let us describe the results of the paper in more detail. In Section 2 we consider some pseudo-topologies and functional spaces on direct products of measure spaces. Basic definitions and results from [A] and [ShT] related to operator synthesis for subsets in a direct product X × Y are recalled. It is proved that a subset with a scattered family of X-sections is equivalent to a countable union of rectangles. A consequence which is used later on is that the set of all solutions (x, y) of an equation of the form 1 Section 3 deals with a kind of spectral synthesis in commutative Banach algebrasthe synthesis with respect to Banach modules. The real distinction of this theory from the classical one is that given a module we get a special class of ideals, the annihilators of subsets in the module, and work with them only. Here our aim is to compare the conditions for an element to be synthetic with respect to a module and to admit spectral synthesis in the algebra. We also relate these conditions to spectra and spectral subspaces of the corresponding multiplication operators.In Section 4 the approach is reduced to the case of operator modules over Varopoulos algebras. Let µ, ν be regular measures on compacts X, Y and H 1 , H 2 the corresponding L 2 -spaces. Then the space B(H 1 , H 2 ) of all bounded operators from H 1 to H 2 becomes a V (X, Y )-module with respect to the action (f ⊗ g) · T = M g T M f , where M f , M g are the multiplication operators. It is proved that F ∈ V (X, Y ) admits spectral synthesis iff it is synthetic with respect to all modules of this kind. This allows to obtain results on spectral synthesis in an operator-theoretical way. The follow...

An Elementary Approach to Elementary Operators on B(H)

“…This result from [22] was proved earlier under more restrictive conditions on A, B and X (see [7] and numerous references therein).…”

“…The reader can consider them as exercises in functional analysis, (s)he can also find their solutions in Section 6 of [22].…”

“…Let us list some examples of p-semidiagonal families (see [22] for transparent proofs or references).…”

“…Moreover we hope that such approach makes the subject really elementary and gives the possibility to present the results with ideas of their proofs in a text of reasonable length. The reader can find transparent proofs and various extensions of the main part of our results in [19,22,23].…”

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