Rota's universal operators and invariant subspaces in Hilbert spaces

Abstract

Abstract. A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and commutes with a quasi-nilpotent injective compact operator with dense range. In particular, this new universal operator invites an approach to the Invariant Subspace Problem that uses properties of operators that commute with the universal operator.

Consequences of universality among Toeplitz operators

“…Defining W by W = H 2 SH 2 , the wandering subspace of S, using the Wold decomposition, H 2 = ⊕ ∞ k=0 S k W, the operator S * can be represented as an upper triangular block matrix that has the identity on the super-diagonal. As it was noted in [8], the only compact operator that commutes with the universal operator S * is the zero operator.…”

“…Theorem 8 (see [8]). For φ as in Equation (2) and J as in Equation (3), the operator T * φ is a universal operator for H 2 and the operator W * ψ,J is an injective compact operator that has dense range and commutes with T * φ .…”

“…More recently, in [8] and described in more detail below (Theorem 8), the authors gave an example of a universal operator, also the adjoint of an analytic Toeplitz operator, that commutes with an injective compact operator with dense range. The compact operator in that example is the adjoint of a weighted composition operator.…”

“…In [8], it was shown that there is a universal operator T that commutes with an injective compact operator W with dense range. At this point, we want to recall the specific descriptions of the operators T = T * φ and W = W * ψ,J .…”

“…At this point, we want to recall the specific descriptions of the operators T = T * φ and W = W * ψ,J . The following easy lemma, taken from [8], tells when such operators commute.…”