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In this paper, we study property $(UW_E)$ for hypercyclic and supercyclic operators. Further, we study the stability of variant of Weyl type theorems under compact perturbation for Toeplitz operators on Bergman space. Also, we give some examples of Toeplitz operators satisfying Weyl type theorems on Bergman space and harmonic Bergman space.

In this paper, we discuss k-quasi-m-isometric composition operators and weighted composition operators on directed graphs with one circuit and more than one branching vertex.

We reconsider studies of Toeplitz operators on function spaces (the weighted Bergman space, the generalized derivative Hardy space) and the H-Toeplitz operators on the Bergman space. Past studies have considered the presence or absence of hyponormality or questions of contractivity or expansivity; we provide structure theorems for these operators that allow us to recapture, and often considerably improve, these results. In some cases these operators or their adjoints are actually in more restrictive classes, such as subnormal or moment infinitely divisible ($\mathcal{MID}$).

In this paper we attempt to lay the foundations for a theory encompassing some natural extensions of the class of subnormal operators, namely the $n$--subnormal operators and the sub-$n$--normal operators. We discuss inclusion relations among the above mentioned classes and other related classes, e.g., $n$--quasinormal and quasi-$n$--normal operators. We show that sub-$n$--normality is stronger than $n$--subnormality, and produce a concrete example of a $3$--subnormal operator which is not sub-$2$--normal. In \cite{CU1}, R.E. Curto, S.H. Lee and J. Yoon proved that if an operator $T$ is subnormal, left-invertible, and such that $T^n$ is quasinormal for some $n \le 2$, then $T$ is quasinormal. in \cite{JS}, P.Pietrzycki and J. Stochel improved this result by removing the assumption of left invertibility. In this paper we consider suitable analogs of this result for the case of operators in the above-mentioned classes. In particular, we prove that the weight sequence of an $n$--quasinormal unilateral weighted shift must be periodic with period at most $n$., Comment: This version has minor improvements in the formulation of some mathematical results; also, a few typographical errors have been corrected

In the study of the geometrically regular weighted shifts (GRWS) -- see [5] -- signed power representing measures (which we call Berger-type charges) played an important role. Motivated by their utility in that context, we establish a general theory for Berger-type charges. We give the first result of which we are aware showing that k-hyponormality alone (as opposed to subnormality) yields measure/charge-related information. More precisely, for signed countably atomic measures with a decreasing sequence of atoms we prove that k-hyponormality of the associated shift forces positivity of the densities of the largest k+1 atoms. Further, for certain completely hyperexpansive weighed shifts, we exhibit a Berger-type charge representation, in contrast (but related) to the classical L\'{e}vy-Khinchin representation. We use Berger-type charges to investigate when a non-subnormal GRWS weighted shift may be scaled to become conditionally positive definite, and close with an example indicating a distinction between the study of moment sequences and the study of weighted shifts.

Geometrically regular weighted shifts (in short, GRWS) are those with weights $\alpha (N,D)$ given by $\alpha_n (N,D) = \sqrt{\frac{p^n + N}{p^n + D}}$, where $p > 1$ and $(N,D)$ is fixed in the open unit square $ (-1, 1)\times (-1, 1)$. We study here the zone of pairs $ (M,P)$ for which the weight $\frac{\alpha (N,D) }{ \alpha (M,P) }$ gives rise to a moment infinitely divisible ($ \mathcal {MID}$) or a subnormal weighted shift, and deduce immediately the analogous results for product weights $\alpha (N,D) \alpha (M,P)$, instead of quotients. Useful tools introduced for this study are a pair of partial orders on the GRWS.

We study an operator-valued generalization of the Haagerup inequality for Gromov hyperbolic groups. In 1978, U. Haagerup showed that if $f$ is a function on the free group $\mathbb{F}_r$ which is supported on the $k$-sphere $S_k=\{x\in \mathbb{F}_r:\ell(x)=k\}$, then the operator norm of its left regular representation is bounded by $(k+1)\|f\|_2$. An operator-valued generalization of it was started by U. Haagerup and G. Pisier. One of the most complete form was given by A. Buchholz, where the $\ell^2$-norm in the original inequality was replaced by $k+1$ different matrix norms associated to word decompositions (this type of inequality is also called Khintchine-type inequality). We provide a generalization of Buchholz's result for hyperbolic groups., Comment: 8 pages, comments welcome!

For recursively generated shifts, we provide definitive answers to two outstanding problems in the theory of unilateral weighted shifts: the Subnormality Problem ({\bf SP}) (related to the Aluthge transform) and the Square Root Problem ({\bf SRP}) (which deals with Berger measures of subnormal shifts). We use the Mellin Transform and the theory of exponential polynomials to establish that ({\bf SP}) and ({\bf SRP}) are equivalent if and only if a natural functional equation holds for the canonically associated Mellin transform. For $p$--atomic measures with $p \le 6$, our main result provides a new and simple proof of the above-mentioned equivalence. Subsequently, we obtain an example of a $7$--atomic measure for which the equivalence fails. This provides a negative answer to a problem posed by G.R. Exner in 2009, and to a recent conjecture formulated by R.E. Curto et al in 2019.

We study a general class of weighted shifts whose weights $\alpha$ are given by $\alpha_n = \sqrt{\frac{p^n + N}{p^n + D}}$, where $p > 1$ and $N$ and $D$ are parameters so that $(N,D) \in (-1, 1)\times (-1, 1)$. Some few examples of these shifts have appeared previously, usually as examples in connection with some property related to subnormality. In sectors nicely arranged in the unit square in $(N,D)$, we prove that these geometrically regular weighted shifts exhibit a wide variety of properties: moment infinitely divisible, subnormal, $k$- but not $(k+1)$-hyponormal, or completely hyperexpansive, and with a variety of well-known functions (such as Bernstein functions) interpolating their weights squared or their moment sequences. They provide subshifts of the Bergman shift with geometric, not linear, spacing in the weights which are moment infinitely divisible. This new family of weighted shifts provides a useful addition to the library of shifts with which to explore new definitions and properties.

In this paper we show that every inner divisor of the operator-valued coordinate function, $zI_E$, is a Blaschke-Potapov factor. We also introduce a notion of operator-valued "rational" function and then show that $\Delta$ is two-sided inner and rational if and only if it can be represented as a finite Blaschke-Potapov product; this extends to operator-valued functions the well-known result proved by V.P. Potapov for matrix-valued functions.

We present a new connection between the classical theory of full and truncated moment problems and the theory of partial differential equations, as follows. For the classical heat equation $\partial_t u = \nu \Delta u$, with initial data $u_0 \in\mathcal{S}(\mathbb{R}^n)$, we first compute the moments $s_{\alpha}(t)$ of the unique solution $u \in \mathcal{S}(\mathbb{R}^n)$. These moments are polynomials in the time variable, of degree comparable to $\alpha$, and with coefficients satisfying a recursive relation. This allows us to define the polynomials for any sequence, and prove that they preserve some of the features of the heat kernel. In the case of moment sequences, the polynomials trace a curve (which we call the heat curve) which remains in the moment cone for positive time, but may wander outside the moment cone for negative time. This provides a description of the boundary points of the moment cone which are also moment sequences. \ We also study how the determinacy of a moment sequence behaves along the heat curve. Next, we consider the transport equation $\partial_t u = ax \cdot \nabla u$, and conduct a similar analysis. Along the way we incorporate several illustrating examples.

For sequences $\alpha \equiv \{\alpha_n\}_{n=0}^{\infty}$ of positive real numbers, called weights, we study the weighted shift operators $W_{\alpha}$ having the property of moment infinite divisibility ($\mathcal{MID}$); that is, for any $p > 0$, the Schur power $W_{\alpha}^p$ is subnormal. We first prove that $W_{\alpha}$ is $\mathcal{MID}$ if and only if certain infinite matrices $\log M_{\gamma}(0)$ and $\log M_{\gamma}(1)$ are conditionally positive definite (CPD). Here $\gamma$ is the sequence of moments associated with $\alpha$, $M_{\gamma}(0),M_{\gamma}(1)$ are the canonical Hankel matrices whose positive semi-definiteness determines the subnormality of $W_{\alpha}$, and $\log$ is calculated entry-wise (i.e., in the sense of Schur or Hadamard). Next, we use conditional positive definiteness to establish a new bridge between $k$--hyponormality and $n$--contractivity, which sheds significant new light on how the two well known staircases from hyponormality to subnormality interact. As a consequence, we prove that a contractive weighted shift $W_{\alpha}$ is $\mathcal{MID}$ if and only if for all $p>0$, $M_{\gamma}^p(0)$ and $M_{\gamma}^p(1)$ are CPD.

We consider weighted shift operators having the property of moment infinite divisibility; that is, for any $p > 0$, the shift is subnormal when every weight (equivalently, every moment) is raised to the $p$-th power. By reconsidering sequence conditions for the weights or moments of the shift, we obtain a new characterization for such shifts, and we prove that such shifts are, under mild conditions, robust under a variety of operations and also rigid in certain senses. In particular, a weighted shift whose weight sequence has a limit is moment infinitely divisible if and only if its Aluthge transform is. We also consider back-step extensions, subshifts, and completions.

We study the Reconstruction-of-the-Measure Problem (ROMP) for commuting 2-variable weighted shifts $W_{(\alpha,\beta)}$, when the initial data are given as the Berger measure of the restriction of $W_{(\alpha,\beta)}$ to a canonical invariant subspace, together with the marginal measures for the 0-th row and 0-th column in the weight diagram for $W_{(\alpha,\beta)}$. We prove that the natural necessary conditions are indeed sufficient. When the initial data correspond to a soluble problem, we give a concrete formula for the Berger measure of $W_{(\alpha,\beta)}$. Our strategy is to build on previous results for back-step extensions and one-step extensions. A key new theorem allows us to solve ROMP for two-step extensions. This, in turn, leads to a solution of ROMP for arbitrary canonical invariant subspaces of $\ell^2(\mathbb{Z}_+^2)$.

Given a bounded sequence \omega of positive numbers and its associated unilateral weighted shift W_{\omega} acting on the Hilbert space \ell^2(\mathbb{Z}_+), we consider natural representations of W_{\omega} as a 2-variable weighted shift, acting on \ell^2(\mathbb{Z}_+^2). Alternatively, we seek to examine the various ways in which the sequence \omega can give rise to a 2-variable weight diagram. Our best (and more general) embedding arises from looking at two polynomials p and q nonnegative on a closed interval I in R_+ and the double-indexed moment sequence \{\int p(r)^k q(r)^{\ell} d\sigma(r)\}_{k,\ell \in \mathbb{Z}_+}, where W_{\omega} is assumed to be subnormal with Berger measure \sigma such that \supp \; \sigma \subseteq I; we call such an embedding a (p,q)-embedding of W_{\omega}. We prove that every (p,q)-embedding of a subnormal weighted shift W_{\omega} is (jointly) subnormal, and we explicitly compute its Berger measure. We apply this result to answer three outstanding questions: (i) Can the Bergman shift A_2 be embedded in a subnormal 2-variable spherically isometric weighted shift W_{(\alpha,\beta)}? If so, what is the Berger measure of W_{(\alpha,\beta)}? (ii) Can a contractive subnormal unilateral weighted shift be always embedded in a spherically isometric 2-variable weighted shift? (iii) Does there exist a hyponormal 2-variable weighted shift \Theta(W_{\omega}) (where \Theta(W_{\omega}) denotes the classical embedding of a hyponormal unilateral weighted shift W_{\omega}) such that some integer power of \Theta(W_{\omega}) is not hyponormal? As another application, we find an alternative way to compute the Berger measure of the Agler j-th shift A_{j} (j\geq 2). Our research uses techniques from the theory of disintegration of measures, Riesz functionals, and the functional calculus for the columns of the moment matrix associated to a polynomial embedding.

For an arbitrary commuting $d$--tuple $\bT$ of Hilbert space operators, we fully determine the spectral picture of the generalized spherical Aluthge transform $\dbT$ and we prove that the spectral radius of $\bT$ can be calculated from the norms of the iterates of $\dbT$. \ Let $\bm{T} \equiv (T_1,\cdots,T_d)$ be a commuting $d$--tuple of bounded operators acting on an infinite dimensional separable Hilbert space, let $P:=\sqrt{T_1^*T_1+\cdots+T_d^*T_d}$, and let $$ \left( \begin{array}{c} T_1 \\ \vdots \\ T_d \end{array} \right) = \left( \begin{array}{c} V_1 \\ \vdots \\ V_d \end{array} \right) P $$ be the canonical polar decomposition, with $(V_1,\cdots,V_d)$ a (joint) partial isometry and $$ \bigcap_{i=1}^d \ker T_i=\bigcap_{i=1}^d \ker V_i=\ker P. $$ \medskip For $0 \le t \le 1$, we define the generalized spherical Aluthge transform of $\bm{T}$ by $$ \Delta_t(\bm{T}):=(P^t V_1P^{1-t}, \cdots, P^t V_dP^{1-t}). $$ We also let $\left\|\bm{T}\right\|_2:=\left\|P\right\|$. \ We first determine the spectral picture of $\Delta_t(\bm{T})$ in terms of the spectral picture of $\bm{T}$; in particular, we prove that, for any $0 \le t \le 1$, $\Delta_t(\bm{T})$ and $\bm{T}$ have the same Taylor spectrum, the same Taylor essential spectrum, the same Fredholm index, and the same Harte spectrum. \ We then study the joint spectral radius $r(\bm{T})$, and prove that $r(\bm{T})=\lim_n\left\|\Delta_t^{(n)}(\bm{T})\right\|_2 \,\, (0 < t < 1)$, where $\Delta_t^{(n)}$ denotes the $n$--th iterate of $\Delta_t$. \ For $d=t=1$, we give an example where the above formula fails.

We study jointly quasinormal and spherically quasinormal pairs of commuting operators on Hilbert space, as well as their powers. We first prove that, up to a constant multiple, the only jointly quasinormal $2$-variable weighted shift is the Helton-Howe shift. Second, we show that a left invertible subnormal operator $T$ whose square $T^{2}$ is quasinormal must be quasinormal. Third, we generalize a characterization of quasinormality for subnormal operators in terms of their normal extensions to the case of commuting subnormal $n$-tuples. Fourth, we show that if a $2$-variable weighted shift $W_{\left(\alpha ,\beta \right) }$ and its powers $W_{\left(\alpha ,\beta \right)}^{(2,1)}$ and $W_{\left(\alpha ,\beta \right)}^{(1,2)}$ are all spherically quasinormal, then $W_{\left( \alpha ,\beta \right)}$ may not necessarily be jointly quasinormal. Moreover, it is possible for both $W_{\left(\alpha ,\beta \right)}^{(2,1)}$ and $W_{\left(\alpha ,\beta \right)}^{(1,2)}$ to be spherically quasinormal without $W_{\left(\alpha ,\beta \right)}$ being spherically quasinormal. Finally, we prove that, for $2$-variable weighted shifts, the common fixed points of the toral and spherical Aluthge transforms are jointly quasinormal., Comment: In press

Let $\mathbf{T} \equiv (T_1,\cdots,T_n)$ be a commuting $n$-tuple of operators on a Hilbert space $\mathcal{H}$, and let $T_i \equiv V_i P \; (1 \le i \le n)$ be its canonical joint polar decomposition (i.e., $P:=\sqrt{T_1^*T_1+\cdots+T_n^*T_n}$, $(V_1,\cdots,V_n)$ a joint partial isometry, and $\bigcap_{i=1}^n \ker T_i = \bigcap_{i=1}^n \ker V_i = \ker P)$. \ The spherical Aluthge transform of $\mathbf{T}$ is the (necessarily commuting) $n$-tuple $\hat{\mathbf{T}}:=(\sqrt{P}V_1\sqrt{P},\cdots,\sqrt{P}V_n\sqrt{P})$. \ We prove that $\sigma_T(\hat{\mathbf{T}})=\sigma_T(\mathbf{T})$, where $\sigma_T$ denotes Taylor spectrum. \ We do this in two stages: away from the origin we use tools and techniques from criss-cross commutativity; at the origin we show that the left invertibility of $\mathbf{T}$ or $\hat{\mathbf{T}}$ implies the invertibility of $P$. \ As a consequence, we can readily extend our main result to other spectral systems that rely on the Koszul complex for their definitions., Comment: In press

In this work, we consider "finite bandwidth" reproducing kernel Hilbert spaces which have orthonormal bases of the form $f_n(z)=z^n \prod_{j=1}^J \left( 1 - a_{n}w_j z \right)$, where $w_1 ,w_2, \ldots w_J $ are distinct points on the circle $\mathbb{T}$ and $\{ a_n \}$ is a sequence of complex numbers with limit $1$. We provide general conditions based on a matrix recursion that guarantee such spaces contain a functional multiple of the Hardy space. Then we apply this general method to obtain strong results for finite bandwidth spaces when $\lim_{n\rightarrow \infty} n (1-a_n)=p$. In particular, we show that point evaluation can be extended boundedly to precisely $J$ additional points on $\mathbb{T}$ and we obtain an explicit functional decomposition of these spaces for $p>1/2$ in analogy with a previous result in the tridiagonal case due to Adams and McGuire. We also prove that multiplication by $z$ is a bounded operator on these spaces and that they contain the polynomials., Comment: 19 pages with references

A bounded linear Hilbert space operator $S$ is said to be a $2$-isometry if the operator $S$ and its adjoint $S^*$ satisfy the relation $S^{*2}S^{2} - 2 S^{*}S + I = 0$. In this paper, we study Hilbert space operators having liftings or dilations to $2$-isometries. The adjoint of an operator which admits such liftings is characterized as the restriction of a backward shift on a Hilbert space of vector-valued analytic functions. These results are applied to concave operators (i.e., operators $S$ such that $S^{*2}S^{2} - 2 S^{*}S + I \le 0$) and to operators similar to contractions or isometries. Two types of liftings to $2$-isometries, as well as the extensions induced by them, are constructed and isomorphic minimal liftings are discussed., Comment: 30 pages ; to appear in J. Operator Th

Let $T$ be a bounded linear operator on a Hilbert space $\mathcal{H}$, and let $T \equiv V|T|$ be the polar decomposition of $T$. The mean transform of $T$ is defined by $\widehat{T}:=\frac{1}{2}(V|T|+|T|V)$. In this paper we study the iterates of the mean transform and we define the mean limit of an operator as the limit (in the operator norm) of those iterates. We obtain new estimates for the numerical range and numerical radius of the mean transform in terms of the original operator. For the special class of unilateral weighted shifts we describe the precise relationship between the spectral radius and the mean limit, and obtain some sharp estimates., Comment: 13 pages, Proc. Amer. Math. Soc. (2018)

We present an alternative solution to nonsingular cubic moment problems, using techniques that are expected to be useful for higher-degree truncated moment problems. In particular, we apply the theory of recursively determinate moment matrices to deal with a case of rank-increasing moment matrix extensions.

Let $\mathbf{T}$ be a pair of commuting hyponormal operators satisfying the so-called quasitriangular property $$ \textrm{dim} \; \textrm{ker} \; (\mathbf{T}-\boldsymbol\lambda) \ge \textrm{dim} \; \textrm{ker} \; (\mathbf{T} - {\boldsymbol\lambda})^*), $$ for every $\boldsymbol\lambda$ in the Taylor spectrum $\sigma(\mathbf{T})$ of $\mathbf{T}$. We prove that the Weyl spectrum of $\mathbf{T}$, $\omega(\mathbf{T})$, satisfies the identity $$ \omega(\mathbf{T})=\sigma(\mathbf{T}) \setminus \pi_{00}(\mathbf{T}), $$ where $\pi_{00}(\mathbf{T})$ denotes the set of isolated eigenvalues of finite multiplicity. Our method of proof relies on a (strictly $2$-variable) fact about the topological boundary of the Taylor spectrum; as a result, our proof does not hold for $d$-tuples of commuting hyponormal operators with $d>2$.

We say that a weighted shift $W_\alpha$ with (positive) weight sequence $\alpha: \alpha_0, \alpha_1, \ldots$ is {\it moment infinitely divisible} (MID) if, for every $t > 0$, the shift with weight sequence $\alpha^t: \alpha_0^t, \alpha_1^t, \ldots$ is subnormal. \ Assume that $W_{\alpha}$ is a contraction, i.e., $0 < \alpha_i \le 1$ for all $i \ge 0$. \ We show that such a shift $W_\alpha$ is MID if and only if the sequence $\alpha$ is log completely alternating. \ This enables the recapture or improvement of some previous results proved rather differently. \ We derive in particular new conditions sufficient for subnormality of a weighted shift, and each example contains implicitly an example or family of infinitely divisible Hankel matrices, many of which appear to be new.

We introduce two natural notions of multivariable Aluthge transforms (toral and spherical), and study their basic properties. In the case of 2-variable weighted shifts, we first prove that the toral Aluthge transform does not preserve (joint) hyponormality, in sharp contrast with the 1-variable case. Second, we identify a large class of 2-variable weighted shifts for which hyponormality is preserved under both transforms. Third, we consider whether these Aluthge transforms are norm-continuous. Fourth, we study how the Taylor and Taylor essential spectra of 2-variable weighted shifts behave under the toral and spherical Aluthge transforms; as a special case, we consider the Aluthge transforms of the Drury-Arveson 2-shift. Finally, we briefly discuss the class of spherically quasinormal 2-variable weighted shifts, which are the fixed points for the spherical Aluthge transform.

Anderson's theorem states that if the numerical range W(A) of an n-by-n matrix A is contained in the unit disk and intersects with the unit circle at more than n points, then it coincides with the (closed) unit dissk. An analogue of this result for compact A in an infinite dimensional setting was established by Gau and Wu. We consider here the case of A being the sum of a normal and compact operator., Comment: 5 pages

In this paper, we study matrix functions of bounded type from the viewpoint of describing an interplay between function theory and operator theory. \ We first establish a criterion on the coprime-ness of two singular inner functions and obtain several properties of the Douglas-Shapiro-Shields factorizations of matrix functions of bounded type. \ We propose a new notion of tensored-scalar singularity, and then answer questions on Hankel operators with matrix-valued bounded type symbols. \ We also examine an interpolation problem related to a certain functional equation on matrix functions of bounded type; this can be seen as an extension of the classical Hermite-Fej\' er Interpolation Problem for matrix rational functions. \ We then extend the $H^\infty$-functional calculus to an $\overline{H^\infty}+H^\infty$-functional calculus for the compressions of the shift. \ Next, we consider the subnormality of Toeplitz operators with matrix-valued bounded type symbols and, in particular, the matrix-valued version of Halmos's Problem 5; we then establish a matrix-valued version of Abrahamse's Theorem. \ We also solve a subnormal Toeplitz completion problem of $2\times 2$ partial block Toeplitz matrices. \ Further, we establish a characterization of hyponormal Toeplitz pairs with matrix-valued bounded type symbols, and then derive rank formulae for the self-commutators of hyponormal Toeplitz pairs., Comment: To appear in Memoirs Amer. Math. Soc.; vii+106 pages in preprint form

We introduce two natural notions of Aluthge transforms (toral and spherical) for 2-variable weighted shifts and study their basic properties. Next, we study the class of spherically quasinormal $2$-variable weighted shifts, which are the fixed points for the spherical Aluthge transform. Finally, we briefly discuss the relation between spherically quasinormal and spherically isometric 2-variable weighted shifts., Comment: Accepted for publication by C.R. Acad. Sci. Paris, Ser. I (2016)

A well known result of C. Cowen states that, for a symbol $\varphi \in L^{\infty }, \; \varphi \equiv \bar{f}+g \;\;(f,g\in H^{2})$, the Toeplitz operator $T_{\varphi }$ acting on the Hardy space of the unit circle is hyponormal if and only if $f=c+T_{\bar{h}}g,$ for some $c\in {\mathbb C}$, $h\in H^{\infty }$, $\left\| h\right\| _{\infty}\leq 1.$ \ In this note we consider possible versions of this result in the {\it Bergman} space case. \ Concretely, we consider Toeplitz operators on the Bergman space of the unit disk, with symbols of the form $$\varphi \equiv \alpha z^n+\beta z^m +\gamma \overline z ^p + \delta \overline z ^q,$$ where $\alpha, \beta, \gamma, \delta \in \mathbb{C}$ and $m,n,p,q \in \mathbb{Z}_+$, $m < n$ and $p < q$. \ By letting $T_{\varphi}$ act on vectors of the form $$z^k+c z^{\ell}+d z^r \; \; (k<\ell

This dissertation summarizes my investigations in operator theory during my PhD studies. The first chapter is an introduction to that field of operator theory which was developed by B. Sz.-Nagy and C. Foias, the theory of power-bounded Hilbert space operators. In the second and third chapter I characterize operators which arise from power-bounded operators asymptotically. Chapter 4 is devoted to provide a possible generalization of (the necessity part of) Sz.-Nagy's famous similarity theorem. In Chapter 5 I collected my results concerning the commutant mapping of asymptotically non-vanishing contractions. In the final chapter the reader can find results about cyclic properties of weighted shift operators on directed trees., Comment: 96 pages, 6 chapters, 3 figures. Page 87-89 was written in Hungarian, but it is the same as page 84-86. phd thesis, University of Szeged

Given real numbers $\beta \equiv \beta ^{\left( 4\right) }\colon \beta_{00}$, $\beta _{10}$, $\beta _{01}$, $\beta _{20}$, $\beta _{11}$, $ \beta _{02}$, $\beta _{30}$, $\beta _{21}$, $\beta _{12}$, $\beta _{03}$, $\beta _{40}$, $\beta _{31}$, $\beta _{22}$, $\beta _{13}$, $\beta _{04}$, with $\beta _{00} >0$, the quartic real moment problem for $\beta $ entails finding conditions for the existence of a positive Borel measure $\mu $, supported in $\mathbb{R}^2$, such that $\beta _{ij}=\int s^{i}t^{j}\,d\mu \;\;(0\leq i+j\leq 4) $. Let $\mathcal{M}(2)$ be the 6 x 6 moment matrix for $\beta^{(4)}$, given by $\mathcal{M}(2)_{\mathbf{i},\mathbf{j}}:=\beta_{\mathbf{i}+\mathbf{j}}$, where $\mathbf{i},\mathbf{j} \in \mathbb{Z}^2_+$ and $\left|\mathbf{i}\right|,\left|\mathbf{j}\right|\le 2$. In this note we find concrete representing measures for $\beta^{(4)}$ when $\mathcal{M}(2)$ is nonsingular; moreover, we prove that it is possible to ensure that one such representing measure is 6-atomic.

For the truncated moment problem associated to a complex sequence $\gamma ^{(2n)}=\{\gamma _{ij}\}_{i,j\in Z_{+},i+j \leq 2n}$ to have a representing measure $\mu $, it is necessary for the moment matrix $M(n)$ to be positive semidefinite, and for the algebraic variety $\mathcal{V}_{\gamma}$ to satisfy $\operatorname{rank}\;M(n) \leq \;$ card$\;\mathcal{V}_{\gamma}$ as well as a consistency condition: the Riesz functional vanishes on every polynomial of degree at most $2n$ that vanishes on $\mathcal{V}_{\gamma}$. In previous work with L. Fialkow and M. M\"{o}ller, the first-named author proved that for the extremal case (rank$\;M(n)=$ card$\;\mathcal{V}_{\gamma}$), positivity and consistency are sufficient for the existence of a representing measure. In this paper we solve the truncated moment problem for cubic column relations in $M(3)$ of the form $Z^{3}=itZ+u\bar{Z}$ ($u,t \in \mathbb{R}$); we do this by checking consistency. For $(u,t)$ in the open cone determined by $0 < \left|u\right| < t < 2 \left|u\right|$, we first prove that the algebraic variety has exactly $7$ points and $\operatorname{rank}\;M(3)=7$; we then apply the above mentioned result to obtain a concrete, computable, necessary and sufficient condition for the existence of a representing measure., Comment: 18 pages, 2 figures

A theorem of Bayer and Teichmann implies that if a finite real multisequence \beta = \beta^(2d) has a representing measure, then the associated moment matrix M_d admits positive, recursively generated moment matrix extensions M_(d+1), M_(d+2),... For a bivariate recursively determinate M_d, we show that the existence of positive, recursively generated extensions M_(d+1),...,M_(2d-1) is sufficient for a measure. Examples illustrate that all of these extensions may be required to show that \beta has a measure. We describe in detail a constructive procedure for determining whether such extensions exist. Under mild additional hypotheses, we show that M_d admits an extension M_(d+1) which has many of the properties of a positive, recursively generated extension., Comment: To appear in J. Operator Theory

The Lifting Problem for Commuting Subnormals (LPCS) asks for necessary and sufficient conditions for a pair of subnormal operators on Hilbert space to admit commuting normal extensions. \ We study LPCS within the class of commuting 2-variable weighted shifts $\mathbf{T} \equiv (T_1,T_2)$ with subnormal components $T_1$ and $T_2$, acting on the Hilbert space $\ell ^2(\mathbb{Z}^2_+)$ with canonical orthonormal basis $\{e_{(k_1,k_2)}\}_{k_1,k_2 \geq 0}$ . \ The \textit{core} of a commuting 2-variable weighted shift $\mathbf{T}$, $c(\mathbf{T})$, is the restriction of $\mathbf{T}$ to the invariant subspace generated by all vectors $e_{(k_1,k_2)}$ with $k_1,k_2 \geq 1$; we say that $c(\mathbf{T})$ is of \textit{tensor form} if it is unitarily equivalent to a shift of the form $(I \otimes W_\alpha, W_\beta \otimes I)$, where $W_\alpha$ and $W_\beta$ are subnormal unilateral weighted shifts. \ Given a 2-variable weighted shift $\mathbf{T}$ whose core is of tensor form, we prove that LPCS is solvable for $\mathbf{T}$ if and only if LPCS is solvable for any power $\mathbf{T}^{(m,n)}:=(T^m_1,T^n_2)$ ($m,n\geq 1$). \, Comment: article in press

Multivariable, real-valued functions induce matrix-valued functions defined on the space of d-tuples of n-times-n pairwise-commuting self-adjoint matrices. We examine the geometry of this space of matrices and conclude that the best notion of differentiation of these matrix-valued functions is differentiation along curves. We prove that a C^1 real-valued function always induces a C^1 matrix function and give an explicit formula for the derivative. We also show that every real-valued C^m function defined on an open rectangle in the plane induces a matrix-valued function that can be m-times continuously differentiated along C^m curves., Comment: 20 pages

In this paper, we will compute the moments of the block operators, induced by the generators of the group, in a group von Neumann algebra and we will observe the identically distributedness., Comment: 15 pages