Standard dilations of q-commuting tuples

Santanu Dey

Displaying similar documents to “Standard dilations of q-commuting tuples”

Conditions equivalent to C* independence

Shuilin Jin, Li Xu, Qinghua Jiang, Li Li (2012)

Studia Mathematica

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Let and be mutually commuting unital C* subalgebras of (). It is shown that and are C* independent if and only if for all natural numbers n, m, for all n-tuples A = (A₁, ..., Aₙ) of doubly commuting nonzero operators of and m-tuples B = (B₁, ..., Bₘ) of doubly commuting nonzero operators of , $S p (A, B) = S p (A) \times S p (B)$ , where Sp denotes the joint Taylor spectrum.

Commuting contractive families

Luka Milićević (2015)

Fundamenta Mathematicae

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A family f₁,..., fₙ of operators on a complete metric space X is called contractive if there exists a positive λ < 1 such that for any x,y in X we have $d (f_{i} (x), f_{i} (y)) \leq λ d (x, y)$ for some i. Austin conjectured that any commuting contractive family of operators has a common fixed point, and he proved this for the case of two operators. We show that Austin’s conjecture is true for three operators, provided that λ is sufficiently small.

Spaces with maximal projection constants

Hermann König, Nicole Tomczak-Jaegermann (2003)

Studia Mathematica

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We show that n-dimensional spaces with maximal projection constants exist not only as subspaces of $l_{\infty}$ but also as subspaces of l₁. They are characterized by a rigid set of vector conditions. Nevertheless, we show that, in general, there are many non-isometric spaces with maximal projection constants. Several examples are discussed in detail.

A classification of projectors

Gustavo Corach, Alejandra Maestripieri, Demetrio Stojanoff (2005)

Banach Center Publications

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A positive operator A and a closed subspace of a Hilbert space ℋ are called compatible if there exists a projector Q onto such that AQ = Q*A. Compatibility is shown to depend on the existence of certain decompositions of ℋ and the ranges of A and $A^{1 / 2}$ . It also depends on a certain angle between A() and the orthogonal of .

On $C_{0 \cdot}$ multi-contractions having a regular dilation

Dan Popovici (2005)

Studia Mathematica

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Commuting multi-contractions of class $C_{0 \cdot}$ and having a regular isometric dilation are studied. We prove that a polydisc contraction of class $C_{0 \cdot}$ is the restriction of a backwards multi-shift to an invariant subspace, extending a particular case of a result by R. E. Curto and F.-H. Vasilescu. A new condition on a commuting multi-operator, which is equivalent to the existence of a regular isometric dilation and improves a recent result of A. Olofsson, is obtained as a consequence. ...

On operators with the same local spectra

Aleksandar Torgašev (1998)

Czechoslovak Mathematical Journal

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Let $B (X)$ be the algebra of all bounded linear operators in a complex Banach space $X$ . We consider operators $T_{1}, T_{2} \in B (X)$ satisfying the relation $σ_{T_{1}} (x) = σ_{T_{2}} (x)$ for any vector $x \in X$ , where $σ_{T} (x)$ denotes the local spectrum of $T \in B (X)$ at the point $x \in X$ . We say then that $T_{1}$ and $T_{2}$ have the same local spectra. We prove that then, under some conditions, $T_{1} - T_{2}$ is a quasinilpotent operator, that is $∥ {(T_{1} - T_{2})}^{n} ∥^{1 / n} \to 0$ as $n \to \infty$ . Without these conditions, we describe the operators with the same local spectra only in some particular cases.

Smooth operators in the commutant of a contraction

Pascale Vitse (2003)

Studia Mathematica

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For a completely non-unitary contraction T, some necessary (and, in certain cases, sufficient) conditions are found for the range of the $H^{\infty}$ calculus, $H^{\infty} (T)$ , and the commutant, T’, to contain non-zero compact operators, and for the finite rank operators of T’ to be dense in the set of compact operators of T’. A sufficient condition is given for T’ to contain non-zero operators from the Schatten-von Neumann classes $S_{p}$ .

On the norm-closure of the class of hypercyclic operators

Christoph Schmoeger (1997)

Annales Polonici Mathematici

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Let T be a bounded linear operator acting on a complex, separable, infinite-dimensional Hilbert space and let f: D → ℂ be an analytic function defined on an open set D ⊆ ℂ which contains the spectrum of T. If T is the limit of hypercyclic operators and if f is nonconstant on every connected component of D, then f(T) is the limit of hypercyclic operators if and only if $f (σ_{W} (T)) \cup z \in ℂ : | z | = 1$ is connected, where $σ_{W} (T)$ denotes the Weyl spectrum of T.

Complete Pick positivity and unitary invariance

Angshuman Bhattacharya, Tirthankar Bhattacharyya (2010)

Studia Mathematica

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The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foiaş. Just as a contraction is related to the Szegö kernel $k_{S} (z, w) = {(1 - z w ̅)}^{- 1}$ for |z|,|w| < 1, by means of $(1 / k_{S}) (T, T *) \geq 0$ , we consider an arbitrary open connected domain Ω in ℂⁿ, a complete Pick kernel k on Ω and a tuple T = (T₁, ..., Tₙ) of commuting bounded operators on a complex separable Hilbert space ℋ such that (1/k)(T,T*) ≥ 0. For a complete Pick kernel the 1/k functional calculus makes sense...

Weak-type inequalities for maximal operators acting on Lorentz spaces

Adam Osękowski (2014)

Banach Center Publications

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We prove sharp a priori estimates for the distribution function of the dyadic maximal function ℳ ϕ, when ϕ belongs to the Lorentz space $L^{p, q}$ , 1 < p < ∞, 1 ≤ q < ∞. The approach rests on a precise evaluation of the Bellman function corresponding to the problem. As an application, we establish refined weak-type estimates for the dyadic maximal operator: for p,q as above and r ∈ [1,p], we determine the best constant $C_{p, q, r}$ such that for any $ϕ \in L^{p, q}$ , ${| | ℳ ϕ | |}_{r, \infty} \leq C_{p, q, r} {| | ϕ | |}_{p, q}$ .

A new characterization of Anderson’s inequality in $C_{1}$ -classes

S. Mecheri (2007)

Czechoslovak Mathematical Journal

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Let $ℋ$ be a separable infinite dimensional complex Hilbert space, and let $ℒ (ℋ)$ denote the algebra of all bounded linear operators on $ℋ$ into itself. Let $A = (A_{1}, A_{2}, \dots, A_{n})$ , $B = (B_{1}, B_{2}, \dots, B_{n})$ be $n$ -tuples of operators in $ℒ (ℋ)$ ; we define the elementary operators $Δ_{A, B} ℒ (ℋ) \mapsto ℒ (ℋ)$ by $Δ_{A, B} (X) = \sum_{i = 1}^{n} A_{i} X B_{i} - X .$ In this paper, we characterize the class of pairs of operators $A, B \in ℒ (ℋ)$ satisfying Putnam-Fuglede’s property, i.e, the class of pairs of operators $A, B \in ℒ (ℋ)$ such that $\sum_{i = 1}^{n} B_{i} T A_{i} = T$ implies $\sum_{i = 1}^{n} A_{i}^{*} T B_{i}^{*} = T$ for all $T \in 𝒞_{1} (ℋ)$ (trace class operators). The main result is the equivalence between this property and the...

Operators which commute with convolutions on subspaces of $L_{\infty} (G)$

Anthony To-Ming Lau (1978)

Colloquium Mathematicae

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Sums of commuting operators with maximal regularity

Christian Le Merdy, Arnaud Simard (2001)

Studia Mathematica

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Let Y be a Banach space and let $S \subset L_{p}$ be a subspace of an $L_{p}$ space, for some p ∈ (1,∞). We consider two operators B and C acting on S and Y respectively and satisfying the so-called maximal regularity property. Let ℬ and be their natural extensions to $S (Y) \subset L_{p} (Y)$ . We investigate conditions that imply that ℬ + is closed and has the maximal regularity property. Extending theorems of Lamberton and Weis, we show in particular that this holds if Y is a UMD Banach lattice and $e^{- t B}$ is a positive contraction...

Note on the isomorphism problem for weighted unitary operators associated with a nonsingular automorphism

K. Frączek, M. Wysokińska (2008)

Colloquium Mathematicae

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We give a negative answer to a question put by Nadkarni: Let S be an ergodic, conservative and nonsingular automorphism on $(X ̃,_{X ̃}, m)$ . Consider the associated unitary operators on $L ² (X ̃,_{X ̃}, m)$ given by $U ̃_{S} f = \sqrt (d (m \circ S) / d m) \cdot (f \circ S)$ and $φ \cdot U ̃_{S}$ , where φ is a cocycle of modulus one. Does spectral isomorphism of these two operators imply that φ is a coboundary? To answer it negatively, we give an example which arises from an infinite measure-preserving transformation with countable Lebesgue spectrum.

Completely 1-complemented subspaces of the Schatten spaces $S^{p}$

Jean Roydor (2007)

Banach Center Publications

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We describe the subspaces of $S^{p}$ (1 ≤ p ≠ 2 < ∞) which are the range of a completely contractive projection.

Algebra isomorphisms between standard operator algebras

Thomas Tonev, Aaron Luttman (2009)

Studia Mathematica

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If X and Y are Banach spaces, then subalgebras ⊂ B(X) and ⊂ B(Y), not necessarily unital nor complete, are called standard operator algebras if they contain all finite rank operators on X and Y respectively. The peripheral spectrum of A ∈ is the set $σ_{π} (A) = λ \in σ (A) : | λ | = m a x_{z \in σ (A)} | z |$ of spectral values of A of maximum modulus, and a map φ: → is called peripherally-multiplicative if it satisfies the equation $σ_{π} (φ (A) \circ φ (B)) = σ_{π} (A B)$ for all A,B ∈ . We show that any peripherally-multiplicative and surjective map φ: → , neither assumed to be...

Second derivatives of norms and contractive complementation in vector-valued spaces

Bas Lemmens, Beata Randrianantoanina, Onno van Gaans (2007)

Studia Mathematica

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We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces $ℓ_{p} (X)$ , where X is a Banach space with a 1-unconditional basis and p ∈ (1,2) ∪ (2,∞). If the norm of X is twice continuously differentiable and satisfies certain conditions connecting the norm and the notion of disjointness with respect to the basis, then we prove that every 1-complemented subspace of $ℓ_{p} (X)$ admits a basis of mutually disjoint elements. Moreover, we show that every contractive projection...

Two results on the Dunkl maximal operator

Luc Deleaval (2011)

Studia Mathematica

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In this article, we first improve the scalar maximal theorem for the Dunkl maximal operator by giving some precisions on the behavior of the constants of this theorem for a general reflection group. Next we complete the vector-valued theorem for the Dunkl-type Fefferman-Stein operator in the $ℤ ₂^{d}$ case by establishing a result of exponential integrability corresponding to the case p = +∞.

Problems on averages and lacunary maximal functions

Andreas Seeger, James Wright (2011)

Banach Center Publications

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We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First we obtain an H¹ to $L^{1, \infty}$ bound for lacunary maximal operators under a dimensional assumption on the underlying measure and an assumption on an $L^{p}$ regularity bound for some p > 1. Secondly, we obtain a necessary and sufficient condition for L² boundedness of lacunary maximal operator associated to averages over convex curves in the plane. Finally we prove an $L^{p}$ ...

A note on rare maximal functions

Paul Alton Hagelstein (2003)

Colloquium Mathematicae

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A necessary and sufficient condition is given on the basis of a rare maximal function $M_{l}$ such that $M_{l} f \in L ¹ ([0, 1])$ implies f ∈ L log L([0,1]).

Weighted norm estimates and $L_{p}$ -spectral independence of linear operators

Peer C. Kunstmann, Hendrik Vogt (2007)

Colloquium Mathematicae

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We investigate the $L_{p}$ -spectrum of linear operators defined consistently on $L_{p} (Ω)$ for p₀ ≤ p ≤ p₁, where (Ω,μ) is an arbitrary σ-finite measure space and 1 ≤ p₀ < p₁ ≤ ∞. We prove p-independence of the $L_{p}$ -spectrum assuming weighted norm estimates. The assumptions are formulated in terms of a measurable semi-metric d on (Ω,μ); the balls with respect to this semi-metric are required to satisfy a subexponential volume growth condition. We show how previous results on $L_{p}$ -spectral independence...

Transference and restriction of maximal multiplier operators on Hardy spaces

Zhixin Liu, Shanzhen Lu (1993)

Studia Mathematica

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The aim of this paper is to establish transference and restriction theorems for maximal operators defined by multipliers on the Hardy spaces $H^{p} (ℝ^{n})$ and $H^{p} (^{n})$ , 0 < p ≤ 1, which generalize the results of Kenig-Tomas for the case p > 1. We prove that under a mild regulation condition, an $L^{\infty} (ℝ^{n})$ function m is a maximal multiplier on $H^{p} (ℝ^{n})$ if and only if it is a maximal multiplier on $H^{p} (^{n})$ . As an application, the restriction of maximal multipliers to lower dimensional Hardy spaces is considered. ...

Fermi Golden Rule, Feshbach Method and embedded point spectrum

Jan Dereziński (1998-1999)

Séminaire Équations aux dérivées partielles

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A method to study the embedded point spectrum of self-adjoint operators is described. The method combines the Mourre theory and the Limiting Absorption Principle with the Feshbach Projection Method. A more complete description of this method is contained in a joint paper with V. Jak $\overset{ˇ}{s}$ ić, where it is applied to a study of embedded point spectrum of Pauli-Fierz Hamiltonians.

Unitary equivalence and decompositions of finite systems of closed densely defined operators in Hilbert spaces

Piotr Niemiec

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An ideal of N-tuples of operators is a class invariant with respect to unitary equivalence which contains direct sums of arbitrary collections of its members as well as their (reduced) parts. New decomposition theorems (with respect to ideals) for N-tuples of closed densely defined linear operators acting in a common (arbitrary) Hilbert space are presented. Algebraic and order (with respect to containment) properties of the class $C D D_{N}$ of all unitary equivalence classes of such N-tuples...

The Embeddability of c₀ in Spaces of Operators

Ioana Ghenciu, Paul Lewis (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Results of Emmanuele and Drewnowski are used to study the containment of c₀ in the space $K_{w *} (X *, Y)$ , as well as the complementation of the space $K_{w *} (X *, Y)$ of w*-w compact operators in the space $L_{w *} (X *, Y)$ of w*-w operators from X* to Y.