Operators having commutant endowed with cyclicity-preserving quasiaffinities (Q1772573)

Let \(\mathcal{H}\) be a (nonzero, separable, complex) Hilbert space and let \(\mathcal{L}(\mathcal{H})\) denote the \(C^*\)-algebra of all (bounded) linear operators on \(\mathcal{H}\). Given a subalgebra \(\mathcal{A}\) of \(\mathcal{L}(\mathcal{H})\), containing the identity operator, a nonempty vector set \(\mathcal{G} \subset \mathcal{H}\) is called cyclic for \(\mathcal{A}\) if the set \(\{ Ag: A \in \mathcal{A}, \;g \in \mathcal{G} \}\) span the whole space. The multiplicity of \(\mathcal{A}\), \(\mu (\mathcal{A})\), is the minimum of the cardinalities of the cyclic sets for \(\mathcal{A}\). Given an operator \(T \in \mathcal{L} (\mathcal{H})\), the multiplicity of \(T\) is \(\mu (T) = \mu (\{ p(T): \;p \text{ polynomial} \})\) and the commutant multiplicity of \(T\) is \(\mu' (T)= \mu ( \{ C \in \mathcal{L} (\mathcal{H}): CT=TC \} )\). It is known that \(\mu (T)\) shows great variety and that the commutant is usually cyclic, that is, \(\mu '(T)=1\), though it can also be arbitrary. An operator \(T_1 \in \mathcal{L}(\mathcal{H}_1)\) is a quasiaffine transform of \(T_2 \in \mathcal{L}(\mathcal{H}_2)\), \(T_1 \prec T_2\), if the set \(\{ C \in \mathcal{L}(\mathcal{H}_1,\mathcal{H}_2): CT_1=T_2C \} \) contains a linear transformation that is injective and has dense range. Then \(T_1\), \(T_2\) are called quasisimilar, \(T_1 \sim T_2\), if \(T_1 \prec T_2\) and \(T_2 \prec T_1\). It is known that quasisimilarity preserves the multiplicity of operators. The analogous question for the commutant-multiplicity was posed by D.~A.\ Herrero and is still open. In previous papers, the author has given partial answers to that question. He showed that all quasinormal operators are commutant cyclic and settled the case when \(V\) is an isometry with unilateral shift part of finite multiplicity. The problem whether \(\mu'(T)=1\) whenever \(T\sim V\), with \(V\) an arbitrary isometry, led to a question concerning the range of bounded, analytic, operator-valued functions, which was left open. In the present paper, this question is solved. Furthermore, Herrero's problem is settled for a large class of quasinormal operators, including all isometries.