arXiv subjects

Gelu Popescu

Publications and source records attributed to Gelu Popescu.

48 records · Page 3Linked to original sources

arXiv2007

Noncommutative Berezin transforms and model theory

We initiate the study of a class of noncommutative domains of n-tuples of bounded linear operators on a Hilbert space, which is generated by certain positivity conditions on polynomials in n noncommutative indeterminates. We obtain Fatou type results, functional calculi, and a model theory for n-tuples of operators in these domains. The main role in this study is played by a class of noncommutative Berezin transforms which is introduced in this paper. Our results extend to noncommutative varieties.

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arXiv2007

Operator theory on noncommutative varieties

We develop a dilation theory for row contractions subject to constraints determined by sets of noncommutative polynomials. Under natural conditions on the constraints, we have uniqueness for the minimal dilation. A characteristic function is associated with any (constrained) row contraction. For pure constrained row contraction, we show that the characteristic function is a complete unitary invariant and provide a model. We also show that the curvature invariant and Euler characteristic asssociated with a Hilbert module generated by an arbitrary (resp. commuting) row contraction can be expressed only in terms of the corresponding (resp. constrained) characteristic function. We provide a commutant lifting theorem for pure constrained row contractions and obtain a Nevanlinna-Pick interpolation result in this setting. Our theory includes, in particular, the commutative and q-commutative cases, while the standard noncommutative dilation theory for row contractions serves as a "universal model".

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arXiv2007

Operator theory on noncommutative varieties, II

We show that the constrained characteristic function is a complete unitary invariant for J-constrained completely non-coisometric (c.n.c.) row contractions, where J is a WOT-closed two-sided ideal of the noncommutative analytic Toeplitz algebra. We provide a model for this class of row contractions in terms of the constrained characteristic functions. In particular, we obtain a model theory for q-commuting c.n.c. row contractions.

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arXiv2005

Characteristic functions and joint invariant subspaces

Let T:=[T_1,..., T_n] be an n-tuple of operators on a Hilbert space such that T is a completely non-coisometric row contraction. We establish the existence of a "one-to-one" correspondence between the joint invariant subspaces under T_1,..., T_n, and the regular factorizations of the characteristic function associated with T. In particular, we prove that there is a non-trivial joint invariant subspace under the operators T_1,..., T_n, if and only if there is a non-trivial regular factorization of the characteristic function. We also provide a functional model for the joint invariant subspaces in terms of the regular factorizations of the characteristic function, and prove the existence of joint invariant subspaces for certain classes of n-tuples of operators. We obtain criterions for joint similarity of n-tuples of operators to Cuntz row isometries. In particular, we prove that a completely non-coisometric row contraction T is jointly similar to a Cuntz row isometry if and only if the characteristic function of T is an invertible multi-analytic operator.

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arXiv2005

Multivariable moment problems

In this paper we solve moment problems for Poisson transforms and, more generally, for completely positive linear maps on unital C*-algebras generated by ''universal'' row contractions associated with the free semigroup with n generators.

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arXiv2005

Multivariable Bohr inequalities

Operator-valued multivariable Bohr type inequalities are obtained for: a class of noncommutative holomorphic functions, generalizing the analytic functions on the open unit disc; the noncommutative disc algebra and the noncommutative analytic Toeplitz algebra; a class of noncommutative selfadjoint harmonic functions, generalizing the real-valued harmonic functions on the open unit disc; Cuntz-Toeplitz algebras ; the reduced (resp.full) group C*-algebra of the free group with n generators; a class of analytic functions on the open unit ball of C^n. The classical Bohr inequality is shown to be a consequence of Fejer's inequality for the coefficients of positive trigonometric polynomials and Haagerup-de la Harpe inequality for nilpotent operators. Moreover, we provide an inequality which, for analytic polynomials on the open unit disc, is sharper than Bohr's inequality.

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arXiv2004

Unitary invariants in multivariable operator theory

The problems considered in this paper come as a natural continuation of our program to develop a free analogue of Sz.-Nagy-Foias theory, for row contractions. The paper is structured as follows: Introduction Part I. Unitary invariants for n-tuples of operators 1. Joint numerical radius 2. Euclidean operator radius 3. Joint numerical range and spectrum 4. $ρ$-operator radius Part II. Joint operator radii, inequalities, and applications 5. von Neumann inequalities 6. Constrained von Neumann inequalities 7. Multivariable Haagerup-de la Harpe inequalities 8. Multivariable Fejer inequalities References

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arXiv2003

Entropy and multivariable interpolation

We define a new notion of entropy for operators on Fock spaces and positive definite multi-Toeplitz kernels on free semigroups. This is studied in connection with factorization theorems for (multi-Toeplitz, multi-analytic, etc.) operators on Fock spaces. These results lead to entropy inequalities and entropy formulas for positive definite multi-Toeplitz kernels on free semigroups (resp.multi-Toeplitz operators) and consequences concerning the extreme points of the unit ball of the noncommutative analytic Toeplitz algebra F_n^\infty. We obtain several geometric characterizations of the multivariable central intertwining lifting, a maximum principle, and a permanence principle for the noncommutative commutant lifting theorem. Under certain natural conditions, we find explicit forms for the maximal entropy solution (and its entropy) for this multivariable commutant lifting theorem. All these results are used to solve maximal entropy interpolation problems in several variables. We obtain explicit forms for the maximal entropy solution (as well as its entropy) of the Sarason, Caratheodory-Schur, and Nevanlinna-Pick type interpolation problems for the noncommutative (resp.commutative) analytic Toeplitz algebra F_n^\infty (resp.W_n^\infty)and their tensor products with B(H,K). In particular, we provide explicit forms for the maximal entropy solutions of several interpolation (resp.optimization) problems on the unit ball of C^n.

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arXiv2003

Similarity and ergodic theory of positive linear maps

In this paper we study the operator inequality ϕ(X)\leq X and the operator equation ϕ(X)= X, where ϕis a w^*-continuous positive (resp. completely positive) linear map on B(H). We show that their solutions are in one-to-one correspondence with a class of Poisson transforms on Cuntz-Toeplitz C^*-algebras, if ϕis completely positive. Canonical decompositions, ergodic type theorems, and lifting theorems are obtained and used to provide a complete description of all solutions, when ϕ(I)\leq I. We show that the above-mentioned inequality (resp. equation) and the structure of its solutions have strong implications in connection with representations of Cuntz-Toeplitz C^*-algebras, common invariant subspaces for n-tuples of operators, similarity of positive linear maps, and numerical invariants associated with Hilbert modules over \CF_n^+, the complex free semigroup algebra generated by the free semigroup on n generators.

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arXiv2003

Commutator lifting inequalities and interpolation

In this paper we obtain a multivariable commutator lifting inequality, which extends to several variables a recent result of Foias, Frazho, and Kaashoek. The inequality yields a multivariable lifting theorem generalizing the noncommutative commutant lifting theorem. This is used to solve new operator-valued interpolation problems of Schur-Caratheodory, Nevanlinna-Pick, and Sarason type on Fock spaces. Some consequences to norm constrained analytic interpolation in the unit ball of C^n are also considered.

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arXiv1997

Noncommutative Interpolation and Poisson transforms

General results of interpolation (eg. Nevanlinna-Pick) by elements in the noncommutative analytic Toeplitz algebra $F^\infty$ (resp. noncommutative disc algebra $A_n$) with consequences to the interpolation by bounded operator-valued analytic functions in the unit ball of ${\bf C}^n$ are obtained. Non-commutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric representations of the quotient algebra $F^\infty/J$ on Hilbert spaces, where $J$ is any $w^*$-closed, 2-sided ideal of $F^\infty$, are obtained and used to construct a $w^*$-continuous, $F^\infty/J$--functional calculus associated to row contractions $T=[T_1,\dots, T_n]$ when $f(T_1,\dots,T_n)=0$ for any $f\in J$. Other properties of the dual algebra $F^\infty/J$ are considered.

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arXiv1994

Factorization and Reflexivity on Fock spaces

The framework of the paper is that of the full Fock space ${\Cal F}^2({\Cal H}_n)$ and the Banach algebra $F^\infty$ which can be viewed as non-commutative analogues of the Hardy spaces $H^2$ and $H^\infty$ respectively. An inner-outer factorization for any element in ${\Cal F}^2({\Cal H}_n)$ as well as characterization of invertible elements in $F^\infty$ are obtained. We also give a complete characterization of invariant subspaces for the left creation operators $S_1,\cdots, S_n$ of ${\Cal F}^2({\Cal H}_n)$. This enables us to show that every weakly (strongly) closed unital subalgebra of $\{φ(S_1,\cdots,S_n):φ\in F^\infty\}$ is reflexive, extending in this way the classical result of Sarason [S]. Some properties of inner and outer functions and many examples are also considered.

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