Strong Operator Topology Research Articles

Norm Transitive Semigroups on Operator Algebra

Let \(\mathcal A \) be a semigroup of bounded linear operators on the Banach algebra \(B(X)\) for a separable Banach space \(X\). We show the transitivity of \(\mathcal A \) with the operator norm topology, implies hypercyclicity with the strong operator topology (SOT) while the converse may not be true. As a consequence, SOT-transitive semigroup of left multiplication operators on \(B(X)\) is characterized.

Hilbert von Neumann modules

We introduce a way of regarding Hilbert von Neumann modules as spaces of operators between Hilbert space, not unlike [Skei], but in an apparently much simpler manner and involving far less machinery. We verify that our definition is equivalent to that of [Skei], by verifying the `Riesz lemma' or what is called `self-duality' in [Skei]. An advantage with our approach is that we can totally side-step the need to go through $C^*$-modules and avoid the two stages of completion - first in norm, then in the strong operator topology - involved in the former approach. We establish the analogue of the Stinespring dilation theorem for Hilbert von Neumann bimodules, and we develop our version of `internal tensor products' which we refer to as Connes fusion for obvious reasons. In our discussion of examples, we examine the bimodules arising from automorphisms of von Neumann algebras, verify that fusion of bimodules corresponds to composition of automorphisms in this case, and that the isomorphism class of such a bimodule depends only on the inner conjugacy class of the automorphism. We also relate Jones' basic construction to the Stinespring dilation associated to the conditional expectation onto a finite-index inclusion (by invoking the uniqueness assertion regarding the latter).

Lie group-Lie algebra correspondences of unitary groups in finite von Neumann algebras

We give an affirmative answer to the question whether there exist Lie algebras for suitable closed subgroups of the unitary group U($¥mathcal{H}$) in a Hilbert space $¥mathcal{H}$ with U($¥mathcal{H}$) equipped with the strong operator topology. More precisely, for any strongly closed subgroup G of the unitary group U($¥mathfrak{M}$) in a finite von Neumann algebra $¥mathfrak{M}$, we show that the set of all generators of strongly continuous one-parameter subgroups of G forms a complete topological Lie algebra with respect to the strong resolvent topology. We also characterize the algebra $¥overline{¥mathfrak{M}}$ of all densely defined closed operators affiliated with $¥mathfrak{M}$ from the viewpoint of a tensor category.

The order topology on the projection lattice of a Hilbert space

Let L(H) denote the complete lattice of projections on a Hilbert space H. On L(H), besides the restriction of the norm and the strong operator topologies (denoted by τu and τs, respectively) one can consider the order topology τo. In Palko (1995) [10] the topologies τo, τs and τu are compared and it is asked whether τs=τu∩τo. Apart from answering this question, showing that τs and τu∩τo are in general different, this paper contributes to the further understanding of the order topology τo and its relation with τs and τu. It is shown that if H is separable and B is a block, i.e. a maximal Boolean sublattice, of L(H), then the restrictions of τs and τu∩τo to B are equal. We also show if (Pi) is a sequence of compact projections, then Pi→0 w.r.t. τs if and only if Pi→0 w.r.t. τo.

Asymptotic behaviour in the set of nonhomogeneous chains of stochastic operators

We study different types of asymptotic behaviour in the set of (infinite dimensional) nonhomogeneous chains of stochastic operators acting on L1(μ) spaces. In order to examine its structure we consider different norm and strong operator topologies. To describe the nature of the set of nonhomogeneous chains of Markov operators with a particular limit behaviour we use the category theorem of Baire. We show that the geometric structure of the set of those stochastic operators which have asymptotically stationary density differs depending on the considered topologies.

A Spectral Condition for Strong Compactness

An algebra of bounded linear operators on a Banach space is said to be strongly compact if its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Banach space is said to be strongly compact if the algebra with identity generated by the operator is strongly compact. Our interest in this notion stems from the work of Lomonosov on the existence of invariant subspaces. A spectral condition is provided for an operator to be strongly compact and for its commutant to be a strongly compact algebra. This condition is applied to test strong compactness for some classes of operators, namely, bilateral weighted shifts, Cesaro operators, and composition operators. Some tools from function theory like the Mellin transform, the principle of analytic continuation, and the Weierstrass approximation theorem arise naturally in the pursuit of this program.

Common hypercyclic vectors for the unitary orbit of a hypercyclic operator

For a separable, infinite dimensional Hilbert space, it was recently shown by the authors that the similarity orbit of a hypercyclic operator contains a path of operators which is dense in the operator algebra with the strong operator topology, and yet the set of common hypercyclic vectors for the entire path is a dense G δ set. Motivated by that result, we show in the present paper that the unitary orbit of any hypercyclic operator contains a path of operators whose closure contains the entire unitary orbit with the strong operator topology, and yet every nonzero vector in the linear span of the orbit of a given hypercyclic vector is a common hypercyclic vector for the entire path.

On a strong metric on the space of copulas and its induced dependence measure

Using the one-to-one correspondence between copulas and Markov operators on L 1 ( [ 0 , 1 ] ) and expressing the Markov operators in terms of regular conditional distributions (Markov kernels) allows to define a metric D 1 on the space of copulas C that is a metrization of the strong operator topology of the corresponding Markov operators. It is shown that the resulting metric space ( C , D 1 ) is complete and separable and that the induced dependence measure ζ 1 , defined as a scalar times the D 1 -distance to the product copula Π, has various good properties. In particular the class of copulas that have maximum D 1 -distance to the product copula is exactly the class of completely dependent copulas, i.e. copulas induced by Lebesgue-measure preserving transformations on [ 0 , 1 ] . Hence, in contrast to the uniform distance d ∞ , Π cannot be approximated arbitrarily well by completely dependent copulas with respect to D 1 . The interrelation between D 1 and the so-called ∂-convergence by Mikusinski and Taylor as well as the interrelation between ζ 1 and the mutual dependence measure ω by Siburg and Stoimenov is analyzed. ζ 1 is calculated for some well-known parametric families of copulas and an application to singular copulas induced by certain Iterated Functions Systems is given.

The singular coupling limit for a simple pure number process

We consider the strongly continuous unitary one-parameter group on , which is related to the pure number quantum stochastic process restricted to the one-particle subspace . We approximate the Hamiltonian of the group by and go with g to . Then, the approximating unitary group converges to in the strong operator topology with a parameter given explicitly.

The Conjugate Class of a Supercyclic Operator

In this paper, we show that the conjugate set of any supercyclic operator T on a separable, infinite dimensional Banach space X contains a path of supercyclic operators which is dense with the strong operator topology, and the set of common supercyclic vectors for the path is a dense Gδ set if σp(T*) is empty.

Locating subsets of B(H) relative to seminorms inducing the strong-operator topology

Let H be a Hilbert space, andA an inhabited, bounded, convex subset ofB(H). We give a constructive proof thatA is weak-operator totally bounded if and only if it is located relative to a certain family of seminorms that induces the strong-operator topology onB(H).

A "typical" contraction is unitary

We show that the set of unitary operators on a separable infinite-dimensional Hilbert space is residual (for the weak operator topology) in the set of all contractions. The same holds for unitary operators in the set of all isometries and with respect to the strong operator topology. The continuous versions are discussed as well. These results are applied to the problem of embedding operators into strongly continuous semigroups.

Common hypercyclic vectors for the conjugate class of a hypercyclic operator

Given a separable, infinite dimensional Hilbert space, it was recently shown by the authors that there is a path of chaotic operators, which is dense in the operator algebra with the strong operator topology, and along which every operator has the exact same dense G δ set of hypercyclic vectors. In the present work, we show that the conjugate set of any hypercyclic operator on a separable, infinite dimensional Banach space always contains a path of operators which is dense with the strong operator topology, and yet the set of common hypercyclic vectors for the entire path is a dense G δ set. As a corollary, the hypercyclic operators on such a Banach space form a connected subset of the operator algebra with the strong operator topology.

Errata to ‘‘Vector measures and the strong operator topology”

Typographical errors which changed important definitions in the paper Vector measures and the strong operator topology (Proc. Amer. Math. Soc. 137 (2009), 2345―2350) are addressed.

On the limit of square-Cesàro means of contractions on Hilbert spaces

By combining tools from functional analysis and algebraic number theory, we investigate the qualitative properties of the square-Cesaro means \({{\frac{1}{N}}\sum_{n=1}^N T^{n^2}}\), where T is a contraction on a Hilbert space. The existence of their limit in the strong operator topology as N → ∞ is known even for general polynomial sequences, but the limit itself has not yet been studied apart from the linear case solved by the von Neumann Ergodic Theorem. The case of the sequence n2 is the next step beyond this linear case, and we show that even though the limit operator is generally not a projection, it still has a relatively simple and interesting structure.

All-derivable points in nest algebras

Suppose that A is an operator algebra on a Hilbert space H . An element V in A is called an all-derivable point of A for the strong operator topology if every strong operator topology continuous derivable mapping φ at V is a derivation. Let N be a complete nest on a complex and separable Hilbert space H . Suppose that M belongs to N with { 0 } ≠ M ≠ H and write M ^ for M or M ⊥ . Our main result is: for any Ω ∈ alg N with Ω = P ( M ^ ) Ω P ( M ^ ) , if Ω | M ^ is invertible in alg N M ^ , then Ω is an all-derivable point in alg N for the strong operator topology.

Fixed point-free isometric actions of topological groups on Banach spaces

We show that every non-precompact topological group admits a fixed point-free continuous action by affine isometries on a suitable Banach space. Thus, precompact groups are defined by the fixed point property for affine isometric actions on Banach spaces. For separable topological groups, in the above statements it is enough to consider affine actions on one particular Banach space: the unique Banach space envelope $\langle\mathbb U\rangle$ of the universal Urysohn metric space $\mathbb U$, known as the Holmes space. At the same time, we show that Polish groups need not admit topologically proper (in particular, free) affine isometric actions on Banach spaces (nor even on complete metric spaces): this is the case for the unitary group $U(\ell^2)$ with strong operator topology, the infinite symmetric group $S_\infty$, etc.

Defect indices of powers of a contraction

Let A be a contraction on a Hilbert space H . The defect index d A of A is, by definition, the dimension of the closure of the range of I - A ∗ A . We prove that (1) d A n ⩽ nd A for all n ⩾ 0 , (2) if, in addition, A n converges to 0 in the strong operator topology and d A = 1 , then d A n = n for all finite n , 0 ⩽ n ⩽ dim H , and (3) d A = d A ∗ implies d A n = d A n ∗ for all n ⩾ 0 . The norm-one index k A of A is defined as sup { n ⩾ 0 : ‖ A n ‖ = 1 } . When dim H = m < ∞ , a lower bound for k A was obtained before: k A ⩾ ( m / d A ) - 1 . We show that the equality holds if and only if either A is unitary or the eigenvalues of A are all in the open unit disc, d A divides m and d A n = nd A for all n , 1 ⩽ n ⩽ m / d A . We also consider the defect index of f ( A ) for a finite Blaschke product f and show that d f ( A ) = d A n , where n is the number of zeros of f .

On the Iterated Duggal Transforms

【For a bounded operator T = $U{\mid}T{\mid}$ (polar decomposition), we consider a transform b $\widehat{T}$ = ${\mid ${\mid}T{\mid}U$ and discuss the convergence of iterated transform of $\widehat{T}$ under the strong operator topology. We prove that such iteration of quasiaffine hyponormal operator converges to a normal operator under the strong operator topology.】

The entangled ergodic theorem in the almost periodic case

Let U be a unitary operator acting on the Hilbert space H , and α : { 1 , … , 2 k } ↦ { 1 , … , k } a pair partition. Then the ergodic average 1 N k ∑ n 1 , … , n k = 0 N - 1 U n α ( 1 ) A 1 U n α ( 2 ) ⋯ U n α ( 2 k - 1 ) A 2 k - 1 U n α ( 2 k ) converges in the strong operator topology provided U is almost periodic, that is when H is generated by the eigenvalues of U . We apply the present result to obtain the convergence of the Cesaro mean of several multiple correlations.