!Commutant Lifting, Tensor Algebras, and Functional Calculus

Popescu, Gelu

doi:10.1017/s0013091598001059

Cited by 16 publications

(24 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Section 1, we present some results concerning constrained Poisson transforms associated with J-constrained row contractions. More about noncommutative Poisson kernels and Poisson transforms on C * -algebras generated by isometries can be found in [17], [2], [18], [19], [21], and [22].…”

Section: Introductionmentioning

confidence: 99%

Operator theory on noncommutative varieties, II

Popescu

2007

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. An n-tuple of operators T := [T 1 , . . . , T n ] on a Hilbert space H is called a J-constrained row contraction ifwhere J is a WOT-closed two-sided ideal of the noncommutative analytic Toeplitz algebra F ∞ n and f (T 1 , . . . , T n ) is defined using the F ∞ n -functional calculus for row contractions.We show that the constrained characteristic function Θ J,T associated with J and T is a complete unitary invariant for J-constrained completely noncoisometric (c.n.c.) row contractions. We also 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.

show abstract

Section: Introductionmentioning

confidence: 99%

Operator theory on noncommutative varieties, II

Popescu

2007

Proc. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…More precisely, we prove that an operator D is in C ≤ (ϕ) Let us mention that, in the particular case when ϕ(I) ≤ I and D := I, the Poisson transform associated with (ϕ, I) was introduced and studied in [37] in connection with a noncommutative von Neumann inequality for row contractions [33]. Several applications of these Poisson transforms were considered in [37], [7], [39], [2], [3], [38], and recently in [9], [10], [40], and [1]. We refer to [4], [27], and [28] for results on completely bounded maps and operator spaces.…”

Section: Introductionmentioning

confidence: 99%

Similarity and ergodic theory of positive linear maps

Popescu¹

2003

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper we study the operator inequality ϕ(X) ≤ X and the operator equation ϕ(X) = X, where ϕ is 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 ϕ is 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) ≤ 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.

show abstract

“…When j = I H and J = I K , we obtain a new proof of the noncommutative commutant lifting theorem [Po4]. We refer to [Po5], [Po7], [Po8], [Po9], and [APo], for applications of this theorem to interpolation in several variables.…”

Section: T N ] Be a Row Contraction With T I ∈ B(h) And Letmentioning

confidence: 99%

“…Let us remark that Theorem 2.1 can be used to obtain versions of NevanlinnaPick type interpolation for F ∞ n . Since the approach is similar to [APo], [Po8], we leave this task to the reader. All the results of this paper hold true if we allow n = ∞.…”

Section: Theorem 21 Let J ∈ B(k) Be a Symmetry On A Hilbert Space Kmentioning

confidence: 99%