Operators onH Research Articles

A noncommutative Gretsky–Ostroy theorem and its applications

Let H \mathcal {H} be a separable Hilbert space and let B ( H ) B(\mathcal {H}) be the ∗ * -algebra of all bounded linear operators on H \mathcal {H} . In the present paper, we prove that a positive/regular operator from L 1 ( 0 , 1 ) L_1(0,1) into an arbitrary separable operator ideal in B ( H ) B(\mathcal {H}) is necessarily Dunford–Pettis, extending and strengthening results due to Gretsky and Ostroy [Glasgow Math. J. 28 (1986), pp. 113–114], and Holub [Proc. Amer. Math. Soc. 104 (1988), pp. 89–95]. Consequently, for an arbitrary atomless von Neumann algebra M \mathcal {M} and an arbitrary KB-ideal C E C_E in B ( H ) B(\mathcal {H}) , the predual M ∗ \mathcal {M}_* of M \mathcal {M} is not isomorphic to any subspace of C E C_E . This observation complements several earlier results.

Characterizations of Double Commutant Property on B H

Let H be a complex Hilbert space. Denote by B H the algebra of all bounded linear operators on H . In this paper, we investigate the non-self-adjoint subalgebras of B H of the form T + B , where B is a block-closed bimodule over a masa and T is a subalgebra of the masa. We establish a sufficient and necessary condition such that the subalgebras of the form T + B has the double commutant property in some particular cases.

Finite dimensional semigroups of unitary endomorphisms of standard subspaces

Let V \mathtt {V} be a standard subspace in the complex Hilbert space H \mathcal {H} and G G be a finite dimensional Lie group of unitary and antiunitary operators on H \mathcal {H} containing the modular group ( Δ V i t ) t ∈ R (\Delta _{\mathtt {V}}^{it})_{t \in \mathbb {R}} of V \mathtt {V} and the corresponding modular conjugation J V J_{\mathtt {V}} . We study the semigroup \[ S V = { g ∈ G ∩ U ⁡ ( H ) : g V ⊆ V } S_{\mathtt {V}} = \{ g\in G \cap \operatorname {U}(\mathcal {H})\colon g\mathtt {V} \subseteq \mathtt {V}\} \] and determine its Lie wedge L ⁡ ( S V ) = { x ∈ g : exp ⁡ ( R + x ) ⊆ S V } \operatorname {\textbf {L}}(S_{\mathtt {V}}) = \{ x \in \mathfrak {g} \colon \exp (\mathbb {R}_+ x) \subseteq S_{\mathtt {V}}\} , i.e., the generators of its one-parameter subsemigroups in the Lie algebra g \mathfrak {g} of G G . The semigroup S V S_{\mathtt {V}} is analyzed in terms of antiunitary representations and their analytic extension to semigroups of the form G exp ⁡ ( i C ) G \exp (iC) , where C ⊆ g C \subseteq \mathfrak {g} is an Ad ⁡ ( G ) \operatorname {Ad}(G) -invariant closed convex cone. Our main results assert that the Lie wedge L ⁡ ( S V ) \operatorname {\textbf {L}}(S_{\mathtt {V}}) spans a 3 3 -graded Lie subalgebra in which it can be described explicitly in terms of the involution τ \tau of g \mathfrak {g} induced by J V J_{\mathtt {V}} , the generator h ∈ g τ h \in \mathfrak {g}^\tau of the modular group, and the positive cone of the corresponding representation. We also derive some global information on the semigroup S V S_{\mathtt {V}} itself.

Paranormal Measurable Operators Affiliated with a Semifinite von Neumann Algebra

Let M be a von Neumann algebra of operators on a Hilbert space H, τ be a faithful normal semifinite trace on M. We define two (closed in the topology of convergence in measure τ) classes P1 and P2 of τ-measurable operators and investigate their properties. The class P2 contains P1. If a τ-measurable operator T is hyponormal, then T lies in P1; if an operator T lies in Pk, then UTU* belongs to Pk for all isometries U from Mand k = 1, 2; if an operator T from P1 admits the bounded inverse T−1 then T−1 lies in P1. If a bounded operator T lies in P1 then T is normaloid, Tn belongs to P1 and a rearrangement μt(Tn) ≥ μt(T )n for all t > 0 and natural n. If a τ-measurable operator T is hyponormal and Tn is τ-compact operator for some natural number n then T is both normal and τ-compact. If an operator T lies in P1 then T 2 belongs to P1. If M= B(H) and τ = tr, then the class P1 coincides with the set of all paranormal operators onH. If a τ-measurable operator A is q-hyponormal (1 ≥ q > 0) and |A*| ≥ μ∞(A)I then Ais normal. In particular, every τ-compact q-hyponormal (or q-cohyponormal) operator is normal. Consider a τ-measurable nilpotent operator Z ≠ 0 and numbers a, b ∈ R. Then an operator Z*Z − ZZ* + aRZ + bSZ cannot be nonpositive or nonnegative. Hence a τ-measurable hyponormal operator Z ≠ 0 cannot be nilpotent.

On the gain of entanglement assistance in the classical capacity of quantum Gaussian channels

Quantum channels are trace-preserving completely positive linear maps between Banach spaces of trace-class operators (Schatten classes of order 1); these are noncommutative analogs of Markov operators in classical probability theory. They also play the role of dynamical maps in quantum theory [1, Chap. 6]. The main characteristics determining the information properties of a quantum channel include its classical entanglement-assisted and unassisted capacities. The classical (unassisted) capacity C(Φ) of a channel Φ determines the limit rate of classical information transmission through Φ with any block coding at the input and the corresponding measurement at the output, and the classical entanglement-assisted channel capacity Cea(Φ) supposes, in addition, the presence of an entangled state between the input and the output of the channel Φ (a detailed description of transmission protocols can be found in [1, Chap. 8]). Since entanglement is an additional resource, it follows that Cea(Φ) ≥ C(Φ) for any channel Φ. LetH be a separable Hilbert space. By T(H) we denote the Banach space of all trace-class operators onH and byS(H), the subset of T(H) consisting of all positive operators with trace 1; we refer to such operators as quantum states and denote them by Greek letters ρ, σ, . . . . We denote a set of quantum states {ρi} with probability distributions {πi} by {πi, ρi} and call it an ensemble of states; the state ρ = ∑ i πiρi is called the average state of the ensemble {πi, ρi}. A quantum channel is a trace-preserving completely positive linear map Φ: T(HA) → T(HB) [1, Chap. 6]. Let H(ρ) be the von Neumann entropy of a state ρ, and let H(ρ‖σ) be the quantum relative entropy of states ρ and σ. For a given channel Φ and any ensemble {πi, ρi} of input quantum states, the output χ-quantity is determined by the expression

Differences of weighted composition operators acting from Bloch space to 𝐻^{∞}

We study the boundedness and compactness of the differences of two weighted composition operators acting from the Bloch space B \mathcal B to the space H ∞ H^\infty of bounded analytic functions on the open unit disk. Such a study has a relationship to the topological structure problem of composition operators on H ∞ H^\infty . Using this relation, we will estimate the operator norms and the essential norms of the differences of two composition operators acting from B \mathcal B to H ∞ H^\infty .

Differences of Composition Operators on the Space of Bounded Analytic Functions in the Polydisc

This paper gives some estimates of the essential norm for the difference of composition operators induced byφandψacting on the space,H∞(Dn), of bounded analytic functions on the unit polydiscDn, whereφandψare holomorphic self-maps ofDn. As a consequence, one obtains conditions in terms of the Carathéodory distance onDnthat characterizes those pairs of holomorphic self-maps of the polydisc for which the difference of two composition operators onH∞(Dn)is compact.

Projections in operator ranges

If H \mathcal {H} is a Hilbert space, A A is a positive bounded linear operator on H \mathcal {H} and S \mathcal {S} is a closed subspace of H \mathcal {H} , the relative position between S \mathcal {S} and A − 1 ( S ⊥ ) A^{-1}(\mathcal {S}^\perp ) establishes a notion of compatibility. We show that the compatibility of ( A , S ) (A,\mathcal {S}) is equivalent to the existence of a convenient orthogonal projection in the operator range R ( A 1 / 2 ) R(A^{1/2}) with its canonical Hilbertian structure.

On the norm of symmetrised two-sided multiplications

The authors provide precise lower bounds for the completely bounded norm of the operatorTa,b: ß(H) → ß(H) defined byTa,b(x) =axb+bxaand the injective norm of the corresponding tensor. Further, they compute the norm of the operatorx↦a*xb+b*xaacting on the space of all conjugate-linear operators onH.

Connected components in the space of composition operators onH∞ functions of many variables

LetE be a complex Banach space with open unit ballBe. The structure of the space of composition operators on the Banach algebra H∞, of bounded analytic functions onBe with the uniform topology, is studied. We prove that the composition operators arising from mappings whose range lies strictly insideBe form a path connected component. WhenE is a Hilbert space or aCo(X)- space, the path connected components are shown to be the open balls of radius 2.

Topological structure of the space of composition operators onH ?

Components and isolated points of the topological space of composition operators onH∞ in the uniform operator topology are characterized. Compact differences of two composition operators are also characterized. With the aid of these results, we show that a component inC(H ∞ ) is not in general the set of all composition operators that differ from the given one by a compact operator.

On the unique finite (SI) decomposition of the Jordan operator ? i=1 n S(?) for certain inner function ?

In this paper, we have proven that for the Jordan blockS(θ) withS(θ) ∈ (SI), ⊕ =1 S(θ) =S(θ) (n) (n ≥ 1) has unique finite (SI) decomposition up to a similarity. As result, we obtain that ifV is a Volterra operator onH=L 2([0, 1]), thenV (n) has unique finite (SI) decomposition.

Weakly compact operators onH ?

We prove that a weakly compact operator fromH∞ or any of its even duals into an arbitrary Banach space is uniformly convexifying. By using this, we establish three dicothomies: (1) every operator defined onH∞ or any of its even duals either fixes a copy ofl∞ or factors through a Banach space having the Banach-Saks property; (2) every quotient ofH∞ or any of its even duals either contains a copy ofl∞ or is super-reflexive; (3) every subspace ofL1/H 0 1 or any of its even duals either contains a complemented copy ofl1 or is super-reflexive.

Diagonalization of compact operators on Hilbert modules overC *-algebras of real rank zero

The classical Hilbert-Schmidt theorem can be extended to compact operators on HilbertA-modules overW*-algebras of finite type; i.e., with minor restrictions, compact operators onH*A can be diagonalized overA. We show that ifB is a weakly denseC*-subalgebra ofA with real rank zero and if some additional condition holds, then the natural extension fromHB toH*A⊃HB of a compact operator can be diagonalized so that the diagonal elements belong to the originalC*-algebraB.

On composition operators onH p -spaces in several variables

In this paper, we prove that a composition operator onH p (B) is Fredholm if and only if it is invertible if and only if its symbol is an automorphism onB, and give the representation of the spectra of a class of composition operators. In addition, using composition operator, we discuss intertwining Toeplitz operators.

Convolution operators onH p (H k n )

LetG=H p (H ) be the (2n+1)-dimensional Heisenberg group over local fieldK. In this paper we prove some theorems about convolution operators onH p (G) and vector-valued Hardy spaces. As an example, the distribution $$\int {_{k^* } {{\varphi dt} \mathord{\left/ {\vphantom {{\varphi dt} {|t|}}} \right. \kern-\nulldelimiterspace} {|t|}}} $$ for some φ∈S(G), ξ φ=0 is a ramified 0-type kernel. These results can be applied to characterizeH p (G) spaces by square functions.

Local spectral theory for invertible composition operators onH p

Invertible composition operators on the Hardy spaceHp have automorphic symbols. For 1<p<∞ andp≠2 it is shown that some elliptic composition operators are scalar while others are generalized scalar but not spectral, that parabolic composition operators are generalized scalar but not spectral and that hyperbolic composition operators do not have the single-valued extension property.

Isometries of certain operator algebras

To a given basis ϕ 1 , … , ϕ n \phi _1,\dotsc ,\phi _n on an n n -dimensional Hilbert space H \mathcal H , we associate the algebra A \mathfrak A of all linear operators on H \mathcal H having every ϕ j \phi _j as an eigenvector. So, A \mathfrak A is commutative, semisimple, and n n -dimensional. Given two algebras of this type, A \mathfrak A and B \mathfrak B , there is a natural algebraic isomorphism τ \tau of A \mathfrak A and B \mathfrak B . We study the question: When does τ \tau preserve the operator norm?

Random rotations of the Wiener path

Let (W, H, μ) be an abstract Wiener space and letR(w) be a strongly measurable random variable with values in the set of isometries onH. Suppose that ∇Rh is smooth in the Sobolev sense and that it is a quasi-nilpotent operator onH for everyh∈H. It is shown that δ(R(w)h) is again a Gaussian (0, |h| 2 )-random variable. Consequently, if (e i ,i∈ℕ)⊂W * is a complete, orthonormal basis ofH, then $$\tilde w = \sum\nolimits_i {(\delta R(w)e_i )e_i } $$ defines a measure preserving transformation, a “rotation”, onW. It is also shown that if for some strongly measurable, operator valued (onH) random variableR, δ(R(w+k)h) is (0, |h| 2 )-Gaussian for allk, h∈H, thenR is an isometry and ∇Rh is quasi-nilpotent for allH∈H. The relation between the stochastic calculi for these Wiener pathsw and $$\tilde w$$ , as well as the conditions of the inverbibility of the map $$w \to \tilde w$$ are discussed and the problem of the absolute continuity of the image of the Wiener measure μ under Euclidean motion on the Wiener space (i.e. $$w \to \tilde w$$ composed with a shift) is studied.

Portraits of frames

We introduce two methods for generating frames of a Hilbert space H \mathcal {H} . The first method uses bounded operators on H \mathcal {H} . The other method uses bounded linear operators on l 2 {l_2} to generate frames of H \mathcal {H} . We characterize all the mappings that transform frames into other frames. We also show how to construct all frames of a given Hilbert space H \mathcal {H} , starting from any given one. We illustrate the results by giving some examples from multiresolution and wavelet theory.