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In this paper, we study hyperrigidity for singly generated unital commutative $C^*$-algebras. We provide four main approaches to hyperrigidity. The first is via operator-moment characterizations of spectral measures, the subject intensively exploited in quantum theory. The second approach is based on dilation theory and is written in terms of the Stone-von Neumann calculus for normal operators. The third, inspired by a recent result of L. G. Brown, concerns the weak and strong convergence of sequences of subnormal (or normal) operators. Finally, the fourth approach deals with the multiplicativity of unital completely positive maps on $C^*$-subalgebras generated by single normal elements. This is inspired by Petz's theorem and its generalizations established first by Arveson in the finite-dimensional case and then by L. G. Brown in general. The above machinery, together with sophisticated operator inequalities (Kadison's, Hansen's and Lieb-Ruskai's), makes it possible to identify certain subsets $G$ of the set $\{t^{*m}t^n\colon m, n = 0,1,2,\ldots\}$ that are hyperrigid in $C^*(t)$, the unital $C^*$-algebra generated by a single normal element $t$. Examples based on the Choquet boundary are included to confirm the optimality of our results.

Kiukas, Lahti and Ylinen asked the following general question. When is a positive operator measure projection valued? A version of this question formulated in terms of operator moments was posed in a recent paper of the present authors. Let $T$ be a selfadjoint operator and $F$ be a Borel semispectral measure on the real line with compact support. For which positive integers $p< q$ do the equalities $T^k =\int_{\mathbb{R}} x^k F(dx)$, $k=p, q$, imply that $F$ is a spectral measure? In the present paper, we completely solve the second problem. The answer is affirmative if $p$ is odd and $q$ is even, and negative otherwise. The case $(p,q)=(1,2)$ closely related to intrinsic noise operator was solved by several authors including Kruszy\'{n}ski and de Muynck as well as Kiukas, Lahti and Ylinen. The counterpart of the second problem concerning the multiplicativity of unital positive linear maps on $C^*$-algebras is also solved.

In a recent paper [9], R. E. Curto, S. H. Lee and J. Yoon asked the following question: Let $T$ be a subnormal operator, and assume that $T^2$ is quasinormal. Does it follow that $T$ is quasinormal?. In [36] we answered this question in the affirmative. In the present paper, we will extend this result in two directions. Namely, we prove that both class A $n$th roots of bounded quasinormal operators and subnormal $n$th roots of unbounded quasinormal operators are quasinormal. We also show that a non-normal quasinormal operator having a quasinormal $n$th root has a non-quasinormal $n$th root.

In the recent paper, R. E. Curto, S. H. Lee, J. Yoon, asked the following question: Let $A$ be a subnormal operator, and assume that $A^2$ is quasinormal. Does it follow that $A$ is quasinormal? In this paper, we give an affirmative answer to this question. In fact, we prove more general result that subnormal $n$th roots of quasinormal operators are quasinormal.

In this paper, two related types of dualities are investigated. The first is the duality between left-invertible operators and the second is the duality between Banach spaces of vector-valued analytic functions. We will examine a pair ($\mathcal{B},\Psi)$ consisting of a reflexive Banach spaces $\mathcal{B}$ of vector-valued analytic functions on which a left-invertible multiplication operator acts and an operator-valued holomorphic function $\Psi$. We prove that there exist a dual pair ($\mathcal{B}^\prime,\Psi^\prime)$ such that the space $\mathcal{B}^\prime$ is unitarily equivalent to the space $\mathcal{B}^*$ and the following intertwining relations hold \begin{equation*} \mathscr{L} \mathcal{U} = \mathcal{U}\mathscr{M}_z^* \quad\text{and}\quad \mathscr{M}_z\mathcal{U} = \mathcal{U} \mathscr{L}^*, \end{equation*} where $\mathcal{U}$ is the unitary operator between $\mathcal{B}^\prime$ and $\mathcal{B}^*$. In addition we show that $\Psi$ and $\Psi^\prime$ are connected through the relation\begin{equation*} \langle(\Psi^\prime( \bar{z}) e_1) (\lambda),e_2\rangle= \langle e_1,(\Psi( \bar{ \lambda}) e_2)(z)\rangle \end{equation*} for every $e_1,e_2\in E$, $z\in \varOmega$, $\lambda\in \varOmega^\prime$. If a left-invertible operator $T$ satisfies certain conditions, then both $T$ and the Cauchy dual operator $T^\prime$ can be modelled as a multiplication operator on reproducing kernel Hilbert spaces of vector-valued analytic functions $\mathscr{H}$ and $\mathscr{H}^\prime$, respectively. We prove that Hilbert space of the dual pair of $(\mathscr{H},\Psi)$ coincide with $\mathscr{H}^\prime$, where $\Psi$ is a certain operator-valued holomorphic function. Moreover, we characterize when the duality between spaces $\mathscr{H}$ and $\mathscr{H}^\prime$ obtained by identifying them with $\mathcal{H}$ is the same as the duality obtained from the Cauchy pairing.

A new concept of a deformed numerical range $W^\rho(T)$ is introduced. Here $T$ is a bounded linear operator or a matrix and $ \rho \in[1,+\infty)$ is a parameter. Each $W^\rho(T)$ is a closed convex set that contains the spectrum of $T$. Furthermore, $W^\rho(T)$ is decreasing with respect to $ \rho $ and $W^2(T)$ coincides with the numerical range. It is also shown that $W^\rho(T)$ is contained in the closed unit disc if and only if $T$ has a $\rho$ unitary dilation in the sense of N\'agy-Foia\c s. The spectral constants of $W^\rho(T)$ are investigated, it is shown that it is monotone and continuous with respect to the parameter $ \rho $.

We introduce generalized multipliers for left-invertible operators which formal Laurent series $U_x(z)=\sum_{n=1}^\infty(P_ET^{n}x) \frac{1}{z^n}+\sum_{n=0}^\infty(P_E{T^{\prime*}}^{n}x)z^n$ actually represent analytic functions on an annulus or a disc., Comment: arXiv admin note: text overlap with arXiv:1808.03339

A new analytic model for left-invertible operators is introduced and investigated. We show that left-invertible operator $T$, which satisfies certain conditions can be modelled as a multiplication operator $\mathscr{M}_z$ on a reproducing kernel Hilbert space of vector-valued analytic functions on an annulus or a disc. A similar result for composition operators in $\ell^2$-spaces is established.

In this paper, we investigate the question of when the equations $A^{*s}A^{s}=(A^{*}A)^{s}$ for all $s \in S$, where $S$ is a finite set of positive integers, imply the quasinormality or normality of $A$. In particular, it is proved that if $S=\{p,m,m+p,n,n+p\}$, where $2\leq m < n$, then $A$ is quasinormal. Moreover, if $A$ is invertible and $S=\{m,n,n+m\}$, where $m \leq n$, then $A$ is normal. Furthermore, the case when $S=\{m,m+n\}$ and $A^{*n}A^n \leq (A^*A)^n$ is discussed.

It is proved that each bounded injective bilateral weighted shift $W$ satisfying the equality $W^{*n}W^{n}=(W^{*}W)^{n}$ for some integer $n\geq 2$ is quasinormal. For any integer $n\geq 2$, an example of a bounded non-quasinormal weighted shift $A$ on a rootless directed tree with one branching vertex which satisfies the equality $A^{*n}A^{n}=(A^{*}A)^{n}$ is constructed. It is also shown that such an example can be constructed in the class of composition operators in $L^2$-spaces over $\sigma$-finite measure spaces., Comment: 11 pages. arXiv admin note: text overlap with arXiv:1310.3144 by other authors