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Noncommutative Berezin kernels are used to obtain Wold decompositions for *-representations of C*-algebras associated with noncommutative polydomains. We study the structure of these representations and C*-algebras and obtain some classification results. [ABSTRACT FROM AUTHOR]

Based on JB*-triple theory, we refine earlier results on the structure of strongly continuous one-parameter semigroups (C0-SGR) of holomorphic Carathéodory isometries of the unit ball in infinite dimensional spin factors, resulting in finite algebraic formulas in terms of joint boundary fixed points, Mobius charts and inverse Laplace transform. [ABSTRACT FROM AUTHOR]

This paper concerns a commutant lifting theorem and a Nevanlinna–Pick type interpolation result in the setting of multipliers from vector-valued Drury–Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in C n . The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball. [ABSTRACT FROM AUTHOR]

Statistical analysis on object data presents many challenges. Basic summaries such as means and variances are difficult to compute. We apply ideas from topology to study object data. We present a framework for using death vectors and persistence landscapes to vectorize object data and perform statistical analysis. We apply this method to some common leaf images that were previously shown to be challenging to compare using a 3D shape techniques. Surprisingly, the most persistent features are shown to be "topological noise" and the statistical analysis depends on the less persistent features which we refer to as the "geometric signal". We also describe the first steps to a new approach to using topology for object data analysis, which applies topology to distributions on object spaces. We introduce a new Fréchet-Morse function technique for probability distribution on a compact object space, extending the Fréchet means lo a larger number of location parameters, including Fréchet antimeans. An example of 3D data analysis to distinguish two flowers using the new location parameters associated with a Veronese-Whitney (VW) embedding of random projective shapes of 3D configurations extracted from a set of pairs of their digital camera images is also given here. [ABSTRACT FROM AUTHOR]

Matrix valued truncated Toeplitz operators act on vector-valued model spaces. They represent a generalization of block Toeplitz matrices. A characterization of these operators analogue to the scalar case is obtained, as well as the determination of the symbols that produce the zero operator. [ABSTRACT FROM AUTHOR]

Several properties of the Harnack domination of linear operators acting on Hilbert space with norm less or equal than one are studied. Thus, the maximal elements for this relation are identified as precisely the singular unitary operators, while the minimal elements are shown to be the isometries and the adjoints of isometries. We also show how a large range of properties (e.g., convergence of iterates, peripheral spectrum, ergodic properties) are transfered from a contraction to one that Harnack dominates it. [ABSTRACT FROM AUTHOR]

The paper discusses a series of results concerning reproducing kernel Hilbert spaces, related to the factorization of their kernels. In particular, it is proved that for a large class of spaces isometric multipliers are trivial. One also gives for certain spaces conditions for obtaining a particular type of dilation, as well as a classification of Brehmer type submodules. [ABSTRACT FROM AUTHOR]

An order relation for contractions on a Hilbert space can be introduced by stating that A ≤ B if and only if A is unitarily equivalent to the restriction of B to an invariant subspace. We discuss the equivalence classes associated to this relation, and identify cases in which they coincide with classes of unitary equivalence. The results extend those for completely nonunitary partial isometries obtained by Garcia, Martin, and Ross. [ABSTRACT FROM AUTHOR]

This paper is an attempt to unify the multivariable operator model theory for ball-like domains and commutative polydiscs and extend it to a more general class of noncommutative polydomains Dqm (H) in B(H)n. An important role in our study is played by noncommutative Berezin transforms associated with the elements of the polydomain. These transforms are used to prove that each such polydomain has a universal model W = {Wi,j} consisting of weighted shifts acting on a tensor product of full Fock spaces. We introduce the noncommutative Hardy algebra F∞(Dqm ) as the weakly closed algebra generated by {Wi,j} and the identity, and use it to provide a WOT-continuous functional calculus for completely non-coisometric tuples in Dqm (H). It is shown that the Berezin transform is a completely isometric isomorphism between F∞(Dqm ) and the algebra of bounded free holomorphic functions on the radial part of Dqm (H). A characterization of the Beurling type joint invariant subspaces under {Wi,j} is also provided. It has been an open problem for quite some time to find significant classes of elements in the commutative polydisc for which a theory of characteristic functions and model theory can be developed along the lines of the Sz.-Nagy-Foias theory of contractions. We give a positive answer to this question, in our more general setting, providing a characterization for the class of tuples of operators in Dqm (H) which admit characteristic functions. The characteristic function is constructed explicitly as an artifact of the noncommutative Berezin kernel associated with the polydomain, and it is proved to be a complete unitary invariant for the class of completely non-coisometric tuples. Using noncommutative Berezin transforms and C*-algebra techniques, we develop a dilation theory on the noncommutative polydomain Dqm (H). [ABSTRACT FROM AUTHOR]

Suppose E is a subset of the unit circle T and H∞ ⊂ L∞ is the Hardy subalgebra. We examine the problem of finding the distance from the characteristic function of E to znH∞. This admits an alternate description as a dual extremal problem. Precise solutions are given in several important cases. The techniques used involve the theory of Toeplitz and Hankel operators as well as the construction of certain conformal mappings. [ABSTRACT FROM AUTHOR]

We investigate truncated Toeplitz operators belonging to the Schatten ideals. We completely characterize such operators when they have an analytic symbol or belong to the ideal of Hilbert-Schmidt operators. We also study model spaces generated by Blaschke products associated with thin sequences, model spaces generated by certain types of singular inner functions, and operators associated with a class of very smooth symbols. [ABSTRACT FROM AUTHOR]

In this paper, certain connections between complex symmetric operators and anti-automorphisms of singly generated C*-algebras are established. This provides a C*-algebra approach to the norm closure problem for complex symmetric operators. For T ∈ B(H) satisfying C*(T)∩K(H) = {0}, we give several characterizations for T to be a norm limit of complex symmetric operators. As applications, we give concrete characterizations for weighted shifts with nonzero weights to be norm limits of complex symmetric operators. In particular, we prove a conjecture of Garcia and Poore. On the other hand, it is proved that an essentially normal operator is a norm limit of complex symmetric operators if and only if it is complex symmetric. We obtain a canonical decomposition for essentially normal operators which are complex symmetric. [ABSTRACT FROM AUTHOR]

We show that the Hardy space on the unit disk is the only nontrivial irreducible reproducing kernel Hilbert space which satisfies the complete Nevanlinna-Pick property and hyponormality of all multiplication operators. [ABSTRACT FROM AUTHOR]

Pick spaces are a class of reproducing kernel Hilbert spaces that generalize the classical Hardy space and the Drury-Arveson reproducing kernel spaces. We give characterizations of certain contractively included subspaces of Pick spaces. These generalize the characterization of closed invariant subspaces of Trent and McCullough, as well as results for the Drury-Arveson space obtained by Ball, Bolotnikov and Fang. [ABSTRACT FROM AUTHOR]

The de Branges-Rovnyak spaces are known to provide an alternate functional model for contractions on a Hilbert space, equivalent to the Sz.-Nagy-Foias model. The scalar de Branges-Rovnyak spaces $${\mathcal{H}(b)}$$ have essentially different properties, according to whether the defining function b is or not extreme in the unit ball of H. For b extreme the model space is just $${\mathcal{H}(b)}$$ , while for b nonextreme an additional construction is required. In the present paper we identify the precise class of contractions which have as a model $${\mathcal{H}(b)}$$ with b nonextreme. [ABSTRACT FROM AUTHOR]

We present two novel results about Hilbert space operators which are nilpotent of order two. First, we prove that such operators are indestructible complex symmetric operators, in the sense that tensoring them with any operator yields a complex symmetric operator. In fact, we prove that this property characterizes nilpotents of order two among all nonzero bounded operators. Second, we establish that every nilpotent of order two is unitarily equivalent to a truncated Toeplitz operator. [ABSTRACT FROM AUTHOR]

We give a complete description of the finite-rank truncated Toeplitz operators. [ABSTRACT FROM AUTHOR]

We provide conditions for composition operators on the Hardy space of the upper half-plane to have a closed range. We also show that, unlike the situation of the unit disk, the operator of composition with an analytic self-map Φ of the upper half-plane can be similar to an isometry even when Φ is far from being an inner function. [ABSTRACT FROM AUTHOR]

Given a strictly increasing sequence $${\Lambda = (\lambda_n)}$$ of nonnegative real numbers, with $${\sum_{n=1}^\infty \frac{1}{\lambda_n}<\infty}$$, the Müntz spaces $${M_\Lambda^p}$$ are defined as the closure in L([0, 1]) of the monomials $${x^{\lambda_n}}$$. We discuss how properties of the embedding $${M_\Lambda^2\subset L^2(\mu)}$$, where μ is a finite positive Borel measure on the interval [0, 1], have immediate consequences for composition operators on $${M^2_\Lambda}$$. We give criteria for composition operators to be bounded, compact, or to belong to the Schatten-von Neumann ideals. [ABSTRACT FROM AUTHOR]

Unlike Toeplitz operators on H², truncated Toeplitz operators do not have a natural matricial characterization. Consequently, these operators are difficult to study numerically. In this paper we provide criteria for a matrix with distinct eigenvalues to be unitarily equivalent to a truncated Toeplitz operator having an analytic symbol. This test is constructive, and we illustrate it with several examples. As a byproduct, we also prove that every complex symmetric operator on a Hilbert space of dimension ≤ 3 is unitarily equivalent to a direct sum of truncated Toeplitz operators. [ABSTRACT FROM AUTHOR]

conjecture of Halmos proved by Choi and Li states that the closure of the numerical range of a contraction on a Hilbert space is the intersection of the closure of the numerical ranges of all its unitary dilations. We show that for C( N) contractions one can restrict the intersection to a smaller family of dilations. This generalizes a finite dimensional result of Gau and Wu. [ABSTRACT FROM AUTHOR]

A multicontraction on a Hilbert space $$\user1{\mathcal{H}}$$ is an n-tuple of operators T = ( T1,..., T n) acting on $$\user1{\mathcal{H}}$$, such that $$\sum\nolimits_{i = 1}^n {T_i T_i^* \, \leq \,1_\user1{\mathcal{H}}}$$. We obtain some results related to the characteristic function of a commuting multicontraction, most notably discussing its behaviour with respect to the action of the analytic automorphisms of the unit ball. [ABSTRACT FROM AUTHOR]

It is known that each normal operator on a Hilbert space with nonempty interior of the spectrum admits vectors with bounded local resolvent. We generalize this result for Banach space operators with the decomposition property (δ) (in particular for decomposable operators). Moreover, the same result holds for operators with interior points in the localizable spectrum. [ABSTRACT FROM AUTHOR]

We study finite rank perturbations of contractions of class C.0 with finite defect indices. The completely nonunitary part of such a perturbation is also of class C.0, while the unitary part is singular. When the defect indices of the original contraction are not equal, it can be shown that almost always (with respect to a suitable measure) the perturbation has no unitary part. [ABSTRACT FROM AUTHOR]

This paper is a continuation of a previous article that appeared in AXIOMS in 2018. A Euler's formula for hyperbolic functions is considered a consequence of a unifying point of view. Then, the unification of Jordan, Lie, and associative algebras is revisited. We also explain that derivations and co-derivations can be unified. Finally, we consider a "modified" Yang–Baxter type equation, which unifies several problems in mathematics. [ABSTRACT FROM AUTHOR]