arXiv subjects

Gelu Popescu

Publications and source records attributed to Gelu Popescu.

At least 19 recordsLinked to original sources

arXiv2024

Noncommutative domains, universal operator models, and operator algebras

Let B(H) be the algebra of all bounded linear operators on a Hilbert space H. The main goal of the paper is to find large classes of noncommutative domains in B(H) with prescribed universal operator models, acting on the full Fock space with n generators, and to study these domains and their universal models in connection with the Hardy algebras and the C^*-algebras they generate. While the class of these domains contains the regular noncommutative domains previously studied in the literature, the main focus of the present paper is on the non-regular domains. The multi-variable operator theory of these domains is developed throughout the paper.

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arXiv2024

Noncommutative domains, universal operator models, and operator algebras, II

In a recent paper, we introduced and studied the class of admissible noncommutative domains $D_{g^{-1}}(H)$ in $B(H)^n$ associated with admissible free holomorphic functions $g$ in noncommutative indeterminates $Z_1,\ldots, Z_n$. Each such a domain admits a universal model ${\bf W}:=(W_1,\ldots, W_n)$ of weighted left creation operators acting on the full Fock space with $n$ generators. In the present paper, we continue the study of these domains and their universal models in connection with the Hardy algebras and the $C^*$-algebras they generate. We obtain a Beurling type characterization of the invariant subspaces of the universal model ${\bf W}:=(W_1,\ldots, W_n)$ and develop a dilation theory for the elements of the noncommutative domain $D_{g^{-1}}(H)$. We also obtain results concerning the commutant lifting and Toeplitz-corrona in our setting as as well as some results on the boundary property for universal models.

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arXiv2024

Noncommutative weighted shifts, joint similarity, and function theory in several variables

The goal of this paper is to study the structure of noncommutative weighted shifts, their properties, and to understand their role as models (up to similarity) for $n$-tuples of operators on Hilbert spaces as well as their implications to function theory on noncommutative (resp.commutative) Reinhardt domains. We obtain a Rota type similarity result concerning the joint similarity of $n$-tuples of operators to parts of noncommutative weighted multi-shifts and provide a noncommutative multivariable analogue of Foias-Pearcy model for quasinilpotent operators. The model noncommutative weighted multi-shift which is studied in this paper is the $n$-tuple $W=(W_1,\ldots, W_n)$, where $W_i$ are weighted left creation operators of the full Fock space with $n$ generators associated with a weight sequence $\boldsymbol μ=\{μ_β\}_{|β|\geq 1}$ of nonnegative numbers. We also represent the injective weighted multi-shifts $W_1,\ldots, W_n$ as ordinary multiplications by $Z_1,\ldots, Z_n$ on a Hilbert space of noncommutative formal power series. This leads naturally to analytic function theory in several complex variables. One of the goal for the remainder of the paper is to analyze the extent to which our noncommutative formal power series represent analytic functions in several noncommutative (resp. commutative) variables and to develop a functional calculus for arbitrary $n$-tuples of operators on a Hilbert space.

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arXiv2020

Functional calculus and multi-analytic models on regular $Λ$-polyballs

The goal of the present paper is to introduce and study noncommutative Hardy spaces associated with the regular $Λ$-polyball, to develop a functional calculus on noncommutative Hardy spaces for the completely non-coisometric (c.n.c.) $k$-tuples in ${\bf B}_Λ(H)$, and to study the characteristic functions and the associated multi-analytic models for the c.n.c. elements in the regular $Λ$-polyball. In addition, we show that the characteristic function is a complete unitary invariant for the class of c.n.c. $k$-tuples in ${\bf B}_Λ(H)$. These results extend the corresponding classical results of Sz.-Nagy--Foia\c s for contractions and the noncommutative versions for row contractions. In the particular case when $n_1=\cdots=n_k=1$ and $Λ_{ij}=1$, we obtain a functional calculus and operator model theory in terms of characteristic functions for $k$-tuples of contractions satisfying Brehmer condition.

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arXiv2020

Brown-Halmos characterization of multi-Toeplitz operators associated with noncommutative poly-hyperballs

We obtain a noncommutative multivariable analogue of Louhichi and Olofsson characterization of Toeplitz operators with harmonic symbols on the weighted Bergman space $A_m({\bf D})$, as well as Eschmeier and Langendorfer extension to the unit ball of ${\bf C}^n$. All our results are proved in the more general setting of noncommutative poly-hyperballs ${\bf D_n^m}(H)$, ${\bf n,m}\in {\bf N}^k$, and are used to characterize the bounded free $k$-pluriharmonic functions with operator coefficients on poly-hyperballs and to solve the associated Dirichlet extension problem. In particular, the results hold for the reproducing kernel Hilbert space with kernel $$ κ_{\bf m}(z,w):=\prod_{i=1}^k \frac{1}{(1-\bar z_i w_i)^{m_i}},\qquad z,w\in {\bf D}^k, $$ where $m_i\geq 1$. This includes the Hardy space, the Bergman space, and the weighted Bergman space over the polydisk.

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arXiv2020

Multi-Toeplitz operators associated with regular polydomains

In this paper we introduce and study the class of weighted multi-Toeplitz operators associated with noncommutative polydomains ${\bf D_f^m}$, ${\bf m}:=(m_1,\ldots, m_k)\in {\bf N}^k$, generated by $k$-tuples ${\bf f}:=(f_1,\ldots, f_k)$ of positive regular free holomorphic functions in a neighborhood of the origin. These operators are acting on the tensor product $F^2(H_{n_1})\otimes \cdots \otimes F^2(H_{n_k})$ of full Fock spaces with $n_i$ generators or, equivalently, they can be viewed as multi-Toeplitz operators acting on tensor products of weighted full Fock spaces. For a large class of polydomains, we show that there are no non-zero compact multi-Toeplitz operators. We characterize the weighted multi-Toeplitz operators in terms of bounded free $k$-pluriharmonic functions on the radial part of ${\bf D_f^m}$ and use the result to obtain an analogue of the Dirichlet extension problem for free $k$-pluriharmonic functions. We show that the weighted multi-Toeplitz operators have noncommutative Fourier representations which can be viewed as noncommutative symbols and can be used to recover the associated operators. We also prove that the weighted multi-Toeplitz operators satisfy a Brown-Halmos type equation associated with the polydomain ${\bf D_f^m}$.

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arXiv2020

Doubly $Λ$-commuting row isometries, universal models, and classification

The goal of the paper is to study the structure of the k-tuples of doubly $Λ$-commuting row isometries and the $C^*$-algebras they generate from the point of view of noncommutative multivariable operator theory. We obtain Wold decompositions, in this setting, and use them to classify the $k$-tuples of doubly $Λ$-commuting row isometries up to a unitary equivalence. We introduce a universal model in this setting, describe its invariant subspaces, and develop a dilation theory on $Λ$-polyballs.

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arXiv2018

Multi-Toeplitz operators and free pluriharmonic functions

We initiate the study of weighted multi-Toeplitz operators associated with noncommutative regular domains in B(H)^n. These operators are acting on the full Fock space with n generators and have as symbols free pluriharmonic functions. Several classical results from complex analysis concerning harmonic functions have analogues in our noncommutative setting. In particular, we show that the bounded free pluriharmonic functions are precisely those which are noncommutative Berezin transforms of weighted multi-Toeplitz operators, and solve the Dirichlet extension problem in this setting. Using noncommutative Cauchy transforms, we provide a free analytic functional calculus for n-tuples of operators, which extends to free pluriharmonic functions.

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arXiv2017

Bohr inequalities for free holomorphic functions on polyballs

Multivariable operator theory is used to provide Bohr inequalities for free holomorphic functions with operator coefficients on the regular polyball. In addition, we obtain analogues of Caratheodory, Fejer, and Egervary-Szazs inequalities for free holomorhic functions with operator coefficients and positive real parts on the polyball. These results are used to provide multivariable analogues of Landau's inequality and Bohr's inequality when the norm is replaced by the numerical radius of an operator.

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arXiv2017

Free Holomorphic Functions on Polydomains

In this paper, we continue to develop the theory of free holomorphic functions on noncommutative regular polydomains. We find analogues of several classical results from complex analysis such as Abel theorem, Hadamard formula, Cauchy inequality, and Liouville theorem for entire functions, in our multivariable setting. We also provide a maximum principle and a Schwarz type lemma. These results are used to prove analogues of Weierstrass, Montel, and Vitali theorems for the algebra of free holomorphic functions on the regular polydomain, which turns out to be a complete metric space.

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arXiv2017

Ando dilations and inequalities on noncommutative varieties

We obtain dilation results which simultaneously generalize Sz.-Nagy dilation theorem for contractions, Ando's dilation theorem for commuting contractions, Sz.-Nagy--Foias commutant lifting theorem, and Schur's representation for the unit ball of H^\infty, in the framework of noncommutative varieties in several variables and Poisson kernels on Fock spaces. This leads to inequalities which, in the particular case of two contractions, are sharper than Ando's inequality and Agler-McCarthy's inequality in the setting of commuting contractive matrices or arbitrary commuting contractions of class C_0. Our results extend to the bi-ball and a large class of noncommutative varieties.

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arXiv2017

Ando dilations and inequalities on noncommutative domains

We obtain intertwining dilation theorems for noncommutative regular domains D_f and noncommutative varieties V_J of n-tuples of operators, which generalize Sarason and Sz.-Nagy--Foias commutant lifting theorem for commuting contractions. We present several applications including a new proof for the commutant lifting theorem for pure elements in the domain D_f (resp. variety V_J) as well as a Schur type representation for the unit ball of the Hardy algebra associated with the variety V_J. We provide Ando type dilations and inequalities for bi-domains D_f \times D_g and bi-varieties V_J \times V_I. In particular, we obtain extensions of Ando's results and Agler-McCarthy's inequality for commuting contractions to larger classes of commuting operators.

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arXiv2017

Hyperbolic geometry on noncommutative polyballs

This paper is an introduction to the hyperbolic geometry of noncommutative polyballs B_n of bounded linear operators on Hilbert spaces. We use the theory of free pluriharmonic functions on polyballs and noncommutative Poisson kernels on tensor products of full Fock spaces to define hyperbolic type metrics on B_n, study their properties, and obtain hyperbolic versions of Schwarz-Pick lemma for free holomorphic functions on polyballs. As a consequence, the polyballs can be viewed as noncommutative hyperbolic spaces. When specialized to the operatorial polydisk D_k, our hyperbolic metric is complete and invariant under the group of all free holomorphic automorphisms of D_k, and the topology induced on D_k is the usual operator norm topology.

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arXiv2016

Free Pluriharmonic Functions on Noncommutative Polyballs

In this paper, we study free k-pluriharmonic functions on noncommutative regular polyballs. These regular polyballs have universal operator models consisting of left creation operators acting on tensor products of full Fock spaces. We introduce and determine the class of kmulti- Toeplitz operators acting on these tensor products and show that the bounded free k-pluriharmonic functions on regular polyballs are precisely the noncommutative Berezin transforms of k-multi-Toeplitz operators. The Dirichlet extension problem on regular polyballs is also solved. It is proved that a free k-pluriharmonic function has continuous extension to the closed polyball if and only if it is the noncommutative Berezin transform of a k-multi-Toeplitz operator in a certain class, which we determine. We provide a Naimark type dilation theorem for direct products of unital free semigroups, and use it to obtain a structure theorem which characterizes the positive free k-pluriharmonic functions on the regular polyball, with operator-valued coefficients. We define the noncommutative the Berezin (resp. Poisson) transform of a completely bounded linear map on the C*-algebra generated by the universal operator model and give necessary an sufficient conditions for a function to be the Poisson transform of a completely bounded (resp. completely positive) map. In the last section of the paper, we obtain Herglotz-Riesz representation theorems for free holomorphic functions on regular polyballs with positive real parts, extending the classical result as well as Korannyi-Pukanszky version in scalar polydisks.

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arXiv2015

Curvature invariant on noncommutative polyballs

In this paper we develop a theory of curvature (resp. multiplicity) invariant for tensor products of full Fock spaces and also for tensor products of symmetric Fock spaces. This is an attempt to find a more general framework for these invariants and extend some of the results obtained by Arveson for the symmetric Fock space, by the author and Kribs for the full Fock space, and by Fang for the Hardy space over the polydisc. To prove the existence of the curvature and its basic properties in these settings requires a new approach based on noncommutative Berezin transforms and multivariable operator theory on polyballs and varieties, as well as summability results for completely positive maps. The results are presented in the more general setting of regular polyballs.

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arXiv2015

Holomorphic automorphisms of noncommutative polyballs

In this paper, we study free holomorphic functions on regular polyballs and provide analogues of several classical results from complex analysis such as: Abel theorem, Hadamard formula, Cauchy inequality, Schwarz lemma, and maximum principle. These results are used together with a class of noncommutative Berezin transforms to obtain a complete description of the group Aut(B_n) of all free holomorphic automorphisms of the polyball. The abstract polyball B_n has a universal model S consisting of left creation operators acting on tensor products of full Fock spaces. We determine: the group of automorphisms of the Cuntz-Toeplitz algebra C*(S) which leaves invariant the noncommutative polyball algebra A_n; the group of unitarily implemented automorphisms of the polyball algebra A_n and the noncommutative Hardy algebra F_n^\infty, respectively. We prove that the free holomorphic automorphism group Aut(B_n) is a sigma-compact, locally compact topological group with respect to the topology induced by an appropriate metric. Finally, we obtain a concrete unitary projective representation of the topological group Aut(B_n)in terms of noncommutative Berezin kernels associated with regular polyballs.

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arXiv2014

Euler characteristic on noncommutative polyballs

In this paper we introduce and study the Euler characteristic associated with algebraic modules generated by arbitrary elements of certain noncommutative polyballs. We provide several asymptotic formulas and prove some of its basic properties. We show that the Euler characteristic is a complete unitary invariant for the finite rank Beurling type invariant subspaces of the tensor product of full Fock spaces $F^2(H_{n_1})\otimes \cdots \otimes F^2(H_{n_k})$, and prove that its range coincides with the interval $[0,\infty)$. We obtain an analogue of Arveson's version of the Gauss-Bonnet-Chern theorem from Riemannian geometry, which connects the curvature to the Euler characteristic. In particular, we prove that if M is an invariant subspace of $F^2(H_{n_1})\otimes \cdots \otimes F^2(H_{n_k})$, $n_i\geq 2$, which is graded (generated by multi-homogeneous polynomials), then the curvature and the Euler characteristic of the orthocomplement of M coincide.

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arXiv2014

Similarity problems in noncommutative polydomains

In this paper we consider several problems of joint similarity to tuples of bounded linear operators in noncommutative polydomains and varieties associated with sets of noncommutative polynomials. We obtain analogues of classical results such as Rota's model theorem for operators with spectral radius less than one, Sz.-Nagy characterization of operators similar to isometries (or unitary operators), and the refinement obtained by Foia\c s and by de Branges and Rovnyak for strongly stable contractions. We also provide analogues of these results in the context of joint similarity of commuting tuples of positive linear maps on the algebra of bounded linear operators on a separable Hilbert space. An important role in this paper is played by a class of noncommutative cones associated with positive linear maps, the Fourier type representation of their elements, and the constrained noncommutative Berezin transforms associated with these elements. It is shown that there is a intimate relation between the similarity problems and the existence of positive invertible elements in these noncommutative cones and the corresponding Berezin kernels.

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