On the Reflexivity of Pairs of Contractions

Kosiek, Marek; Octavio, Alfredo; Ptak, Marek

doi:10.2307/2160724

Cited by 3 publications

(4 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The paper is generally based on the results from [2], [7]- [8], [13], [46]- [47], [56]- [58]; however it also contains some new material.…”

Section: Introductionmentioning

confidence: 99%

On the Existence of Invariant Subspaces and Reflexivity of N-Tuples of Operators

Ptak¹

1997

Taiwanese J. Math.

Self Cite

View full text Add to dashboard Cite

Recent results concerning the existence of a common nontrivial invariant subspace and reflexivity for families of commuting linear bounded Hilbert space operators will be presented; starting with the families of linear transformations on finite dimensional space, through families of isometries, jointly quasinormal operators and spherical isometries, finishing with N -tuples of contractions with dominating spectra. This paper is based On the notes for the series of lectures given in the De-

show abstract

“…The paper is generally based on the results from [2], [7]- [8], [13], [46]- [47], [56]- [58]; however it also contains some new material.…”

Section: Introductionmentioning

confidence: 99%

On the Existence of Invariant Subspaces and Reflexivity of N-Tuples of Operators

Ptak¹

1997

Taiwanese J. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…x − (w) = k 1 (w, w) + k 2 (w, w) + k 1 (w, w) − k 2 (w, w) 2 + 4a 2 |w| 6 k 1 (w, w) + k 2 (w, w) − k 1 (w, w) − k 2 (w, w) 2 + 4a 2 |w| 6 k 1 (w, w) + k 2 (w, w) + |k 1 (w, w) − k 2 (w, w)| + 2a|w| 3 k 1 (w, w) + k 2 (w, w) − |k 1 (w, w) − k 2 (w, w)| − 2a|w| 3 max k 1 (w, w) + a k 2 (w, w) − a , k 2 (w, w) + a k 1 (w, w) − a , which, in view of (33) and (34), is easily seen to be of polynomial order as a function of |w| 2 . This completes the verification of (5).…”

Section: 1mentioning

confidence: 77%

“…For g, h ∈ E, define f g,h (w) = f (w), g E h, w ∈ D. Since {x, y} is an orthogonal basis of E, in order to show that f g,h ∈ H , it suffices to check that f x,y , f y,x , f x,x , f y,y ∈ H . It is easy to see using (32) which, in view of ( 33) and (34), is easily seen to be of polynomial order as a function of |w| 2 . This completes the verification of (5).…”

Section: 1mentioning

confidence: 99%

“…This provides several new classes of examples and recovers special cases of various known results in one and several variables, see [47,Theorem 3], [50, Section 10, Proposition 37], [10,Theorem 15], [12], [33, Theorem 5.2], [49,Theorem 4.2], [37, Section 0, Theorem 4], [4, Theorem A] (cf. [18,Proposition 4.4], [43,Theorem 3], [37,Theorem 3], [34,Theorem 3.1], [26, Corollary 3.7], [7,Theorem 1.2], [27,Theorem 2.11], [30,Corollary 7]). It is worth noting that the techniques employed in the proofs of Theorems 3.1 and 4.1 have some common features (e.g.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Commutants and reflexivity of multiplication tuples on vector-valued reproducing kernel Hilbert spaces

Chavan

Podder

Trivedi

2018

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

Motivated by the theory of weighted shifts on directed trees and its multivariable counterpart, we address the question of identifying commutant and reflexivity of the multiplication d-tuple Mz on a reproducing kernel Hilbert space H of E-valued holomorphic functions on Ω, where E is a separable Hilbert space and Ω is a bounded domain in C d admitting bounded approximation by polynomials. In case E is a finite dimensional cyclic subspace for Mz, under some natural conditions on the B(E)-valued kernel associated with H , the commutant of Mz is shown to be the algebra H ∞ B(E) (Ω) of bounded holomorphic B(E)-valued functions on Ω, provided Mz satisfies the matrix-valued von Neumann's inequality. This generalizes a classical result of Shields and Wallen (the case of dim E = 1 and d = 1). As an application, we determine the commutant of a Bergman shift on a leafless, locally finite, rooted directed tree T of finite branching index. As the second main result of this paper, we show that a multiplication d-tuple Mz on H satisfying the von Neumann's inequality is reflexive. This provides several new classes of examples as well as recovers special cases of various known results in one and several variables. We also exhibit a family of tri-diagonal B(C 2 )-valued kernels for which the associated multiplication operators Mz are non-hyponormal reflexive operators with commutants equal to H ∞ B(C 2 ) (D).2010 Mathematics Subject Classification. Primary 46E22, 47A13, Secondary 46E40, 47B37.

show abstract

Expanding the joint spectrum of pairs of commuting contractions

Ionescu

1996

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We present a connection between solving the invariant subspace problem for a single operator T on Hilbert space and the existence of a common invariant subspace for two commuting related operators. In particular, we reduce the problem of the existence of nontrivial invariant subspaces for a single contraction with spectral radius one to the problem of the existence of common nontrivial invariant subspaces for a pair of commuting contractions with large joint spectra.

show abstract

On the Reflexivity of Pairs of Contractions

Cited by 3 publications

References 1 publication

On the Existence of Invariant Subspaces and Reflexivity of N-Tuples of Operators

On the Existence of Invariant Subspaces and Reflexivity of N-Tuples of Operators

Commutants and reflexivity of multiplication tuples on vector-valued reproducing kernel Hilbert spaces

Expanding the joint spectrum of pairs of commuting contractions

Contact Info

Product

Resources

About