Consequences of universality among Toeplitz operators

Abstract

The Invariant Subspace Problem for Hilbert spaces is a long-standing question and the use of universal operators in the sense of Rota has been one tool for studying the problem. The best known universal operators have been adjoints of analytic Toeplitz operators or unitarily equivalent to them. We present many examples of Toeplitz operators whose adjoints are universal operators and exhibit some of their common properties. Some ways in which the invariant subspaces of these universal operators interact with operators in their commutants are given. Special attention is given to the closed subalgebra, not always the zero algebra, of compact operators in their commutants. Finally, three questions connecting shift invariant subspaces and invariant subspaces of analytic Toeplitz operators are raised. Positive answers for both of the first two imply the existence of non-trivial invariant subspaces for every bounded operator on separable Hilbert spaces of dimension two or more.

A new proof of a Nordgren, Rosenthal and Wintrobe Theorem on universal operators

“…Let us point out that the argument addressed to prove condition (ii) in Caradus Theorem is a particular instance of a more general result: If f is a bounded analytic function in D and there is > 0 so that |f (e iθ )| ≥ almost everywhere on the unit circle, then 1/f is in L ∞ (∂D) and the (non-analytic) Toeplitz operator T 1/f is a left inverse for the analytic Toeplitz operator T f . (See [5,Lemma 3], for instance).…”

On Universal Operators and Universal Pairs

“…Only very recently, an alternative argument for the universality of C ϕ − λI on H 2 (D) was given [8]. For other concrete examples of universal operators, see [5,7,25,26]. Moreover, the connection between the invariant subspace problem and universality has motivated recent work on the lattice of invariant subspaces of C ϕ on H 2 (D) for hyperbolic automorphisms ϕ, e.g.…”

“…In particular, U + K ∈ UC + (H) whenever U ∈ UC(H) and K is a compact operator. In the sequel, we denote by K(H) the closed ideal of L(H) consisting of compact operators on H. Moreover, in[5, Theorem 2], the authors obtained by direct means a perturbation result for the class UC(H) which contains more detailed information. We also recall that the universal model operator B ∞ has the stronger property that its restrictions represent suitable multiples cT up to unitary equivalence for any T ∈ L(H) (e.g [9,.…”

Simetria complexa e universalidade para operadores de Toeplitz do polidisco

“…A relação entre o PSI e operadores universais é a seguinte: Se U é um operador universal para o espaço de Hilbert H, então o PSI é equivalente à afirmação que todo subespaço invariante de dimensão infinita de U contém um subespaço invariante próprio não trivial. Devido a essa relação, na última década, diversos pesquisadores têm dado atenção ao estudo dos operadores universais, a exemplo de [6,7,42,11,12,13,33]. Um operador U L♣Hq é chamado de operador universal para H se para cada operador limitado não nulo A sobre H, existem um subespaço invariante M de U e um escalar não-nulo µ tais que µA é similar a U ⑤ M , ou seja, existe um isomorfismo linear X de H sobre M tal que U X ✏ µXA.…”