The ultimate estimate of the upper norm bound for the summation of operators

Let A and B be bounded linear operators acting on a Hilbert space H. It is shown that the triangular inequality serves as the ultimate estimate of the upper norm bound for the sum of two operators in the sense that

sup{∥U^*AU+V^*BV∥:U and V are unitaries}=min{∥A+μI∥+∥B-μI∥:μ∈C}.     

Convergence of the iterated Aluthge transform sequence for diagonalizable matrices

Given an r×r complex matrix T, if T=U|T| is the polar decomposition of T, then, the Aluthge transform is defined by

Δ(T)=|T|^1/2U|T|^1/2.     

The isometry group of the Urysohn space as a Lévy group

We prove that the isometry group Iso(U) of the universal Urysohn metric space U equipped with the natural Polish topology is a Lévy group in the sense of Gromov and Milman, that is, admits an approximating chain of compact (in fact, finite) subgroups, exhibiting the phenomenon of concentration of measure. This strengthens an earlier result by Vershik stating that Iso(U) has a dense locally finite subgroup.     

A theorem of Hrushovski-Solecki-Vershik applied to uniform and coarse embeddings of the Urysohn metric space

A theorem proved by Hrushovski for graphs and extended by Solecki and Vershik (independently from each other) to metric spaces leads to a stronger version of ultrahomogeneity of the infinite random graph R, the universal Urysohn metric space U, and other related objects. We show how the result can be used to average out uniform and coarse embeddings of U (and its various counterparts) into normed spaces. Sometimes this leads to new embeddings of the same kind that are metric transforms and besides extend to affine representations of various isometry groups. As an application of this technique, we show that U admits neither a uniform nor a coarse embedding into a uniformly convex Banach space.     

Regularity of Skorohod integral processes based on integrands in a finite Wiener chaos

Summary Letf be a square integrable kernel on them-dimensional unit cube,U the Skorohod integral process in them ^th Wiener chaos associated with it. Isoperimetric inequalities for functions on Wiener space yield the exponential integrability of the increments ofU. To this result we apply the majorizing measure technique to show thatU possesses a continuous version and give an upper bound of its modulus of continuity.     

Tychonoff expansions with prescribed resolvability properties

The recent literature offers examples, specific and hand-crafted, of Tychonoff spaces (in ZFC) which respond negatively to these questions, due respectively to Ceder and Pearson (1967) [3] and to Comfort and García-Ferreira (2001) [5]: (1) Is every ω-resolvable space maximally resolvable? (2) Is every maximally resolvable space extraresolvable? Now using the method of KID expansion, the authors show that every suitably restricted Tychonoff topological space (X,T) admits a larger Tychonoff topology (that is, an “expansion”) witnessing such failure. Specifically the authors show in ZFC that if (X,T) is a maximally resolvable Tychonoff space with S(X,T)?Δ(X,T)=κ, then (X,T) has Tychonoff expansions U=Ui (1?i?5), with Δ(X,Ui)=Δ(X,T) and S(X,Ui)?Δ(X,Ui), such that (X,Ui) is: (i=1) ω-resolvable but not maximally resolvable; (i=2) [if κ^′ is regular, with S(X,T)?κ^′?κ] τ-resolvable for all τ<κ^′, but not κ^′-resolvable; (i=3) maximally resolvable, but not extraresolvable; (i=4) extraresolvable, but not maximally resolvable; (i=5) maximally resolvable and extraresolvable, but not strongly extraresolvable.     

Complement preserving operators

An operatorTVV on a real inner product space is called complement preserving if, wheneverU is aT-invariant subspace ofV the orthogonal complementU is alsoT-invariant. In this note we obtain some results on such operators.     

New basic result in classical descriptive set theory: Preservation of completeness

The aim of this note is to prove the following result:Assume that f is a continuous function from the space of irrationals ωω onto Y such that the image f(U) of every set U which is open and closed in ωω is the union of one open and one closed set. Then Y is a completely metrizable space.     

Convexity properties of harmonic measures

It is shown that any convex combination of harmonic measures , where U₁,…,Uk are relatively compact open neighborhoods of a given point x∈Rd, d?2, can be approximated by a sequence of harmonic measures such that each Wn is an open neighborhood of x in U₁∪?∪Uk.This answers a question raised in connection with Jensen measures. Moreover, it implies that, for every Green domain X containing x, the extremal representing measures for x with respect to the convex cone of potentials on X (these measures are obtained by balayage of the Dirac measure at x on Borel subsets of X) are dense in the compact convex set of all representing measures.This is achieved approximating balayage on open sets by balayage on unions of balls which are pairwise disjoint and very small with respect to their mutual distances and then reducing the size of these balls in a suitable manner.These results, which are presented simultaneously for the classical potential theory and for the theory of Riesz potentials, can be sharpened if the complements or the boundaries of the open sets have a capacity doubling property. The methods developed for this purpose (continuous balayage on increasing families of compact sets, approximation using scattered sets with small capacity) finally lead to answers even in a very general potential-theoretic setting covering a wide class of second order partial differential operators (uniformly elliptic or in divergence form, or sums of squares of vector fields satisfying Hörmander's condition, for example, sub-Laplacians on stratified Lie algebras).     

Invariant subspaces for polynomially bounded operators

Let T be a polynomially bounded operator on a Banach space X whose spectrum contains the unit circle. Then T∗ has a nontrivial invariant subspace. In particular, if X is reflexive, then T itself has a nontrivial invariant subspace. This generalizes the well-known result of Brown, Chevreau, and Pearcy for Hilbert space contractions.     

Operators as spectral integrals of operator-valued functions from the study of multivariate stationary stochastic processes

P. Masani and the author have previously answered the question, “When is an operator on a Hilbert space $\text{H}$ the integral of a complex-valued function with respect to a given spectral (projection-valued) measure?” In this paper answers are given to the question, “When is a linear operator from $\text{H}$^q to $\text{H}$^p the integral of a spectral measure?”; here the values of the integrand are linear operators from the square-summable q-tuples of complex numbers to the square-summable p-tuples of complex numbers, and our spectral measure for $\text{H}$^q is the “inflation” of a spectral measure for $\text{H}$. In the course of this paper, we make available tools for handling the spectral analysis of q-variate weakly stationary processes, 1 ≤ q ≤ ∞, which should enable researchers to deal in the future with the case q = ∞. We show as one application of our theory that if U = ∫(in0, 2π]e^?iθE(dθ) is a unitary operator on $\text{H}$ and if T is a bounded linear operator from $\text{H}$^q to $\text{H}$^q (1 ≤ q ≤ ∞) which is a prediction operator for each stationary process (Uⁿx)_?∞^∞ ?$\text{H}$^q (for each x = (xⁱ)_i^j ∈ $\text{H}$^q, Uⁿx = (Uⁿxⁱ)_i=1^q), then T is a spectral integral, ∫(_0,2π)]Φ(θ) E(dθ), and the Banach norm of T, |T|_B = ess sup |Φ(θ)|_B.     

Topological classification of affine operators on unitary and Euclidean spaces

We study affine operators on a unitary or Euclidean space U up to topological conjugacy. An affine operator is a map f:U→U of the form f(x)=Ax+b, in which A:U→U is a linear operator and b∈U. Two affine operators f and g are said to be topologically conjugate if g=h^-1fh for some homeomorphism h:U→U.If an affine operator f(x)=Ax+b has a fixed point, then f is topologically conjugate to its linear part A. The problem of classifying linear operators up to topological conjugacy was studied by Kuiper and Robbin [Topological classification of linear endomorphisms, Invent. Math. 19 (2) (1973) 83-106] and other authors.Let f:U→U be an affine operator without fixed point. We prove that f is topologically conjugate to an affine operator g:U→U such that U is an orthogonal direct sum of g-invariant subspaces V and W,

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the restriction g∣V of g to V is an affine operator that in some orthonormal basis of V has the form

(x₁,x₂,…,x_n)?(x₁+1,x₂,…,x_n-1,εx_n)     

On subgroups of minimal topological groups

A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. The Roelcke uniformity (or lower uniformity) on a topological group is the greatest lower bound of the left and right uniformities. A group is Roelcke-precompact if it is precompact with respect to the Roelcke uniformity. Many naturally arising non-Abelian topological groups are Roelcke-precompact and hence have a natural compactification. We use such compactifications to prove that some groups of isometries are minimal. In particular, if U₁ is the Urysohn universal metric space of diameter 1, the group Iso(U₁) of all self-isometries of U₁ is Roelcke-precompact, topologically simple and minimal. We also show that every topological group is a subgroup of a minimal topologically simple Roelcke-precompact group of the form Iso(M), where M is an appropriate non-separable version of the Urysohn space.     

Three problems on the lengths of increasing runs

Let U₁, U₂,… be a sequence of independent, uniform (0, 1) r.v.'s and let R₁, R₂,… be the lengths of increasing runs of {U_i}, i.e., X₁=R₁=inf{i:U_i+1<U_i},…, X_n=R₁+R₂+?+R_n=inf{i:i>X_n?1,U_i+1<U_i}. The first theorem states that the sequence $\text{(}\text{3}\text{2}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(X}{}_{\mathrm{n}}\text{?2n)}$ can be approximated by a Wiener process in strong sense.Let τ(n) be the largest integer for which R₁+R₂+?+R_τ(n)?n, $\text{R}{}^1{}_{\mathrm{n}}\text{=n?(R}{}_1\text{+R}{}_2\text{+?+R}{}_{\mathrm{\tau (n)}}\text{)}$ and $\text{M}{}_{\mathrm{n}}\text{=}\text{max}\text{{R}{}_1\text{,R}{}_2\text{,…,R}{}_{\mathrm{\tau (n)}}\text{,R}{}^1{}_{\mathrm{n}}\text{}}$. Here M_n is the length of the longest increasing block. A strong theorem is given to characterize the limit behaviour of M_n.The limit distribution of the lengths of increasing runs is our third problem.     

Harmonic measures for a point may form a square

Let X be a Green domain in Rd, d?2, x∈X, and let Mx(P(X)) denote the compact convex set of all representing measures for x. Recently it has been proven that the set of harmonic measures , U open in X, x∈U, which is contained in the set of extreme points of Mx(P(X)), is dense in Mx(P(X)). In this paper, it is shown that Mx(P(X)) is not a simplex (and hence not a Poulsen simplex). This is achieved by constructing open neighborhoods U₀, U₁, U₂, U₃ of x such that the harmonic measures are pairwise different and . In fact, these measures form a square with respect to a natural L²-structure. Since the construction is mainly based on having certain symmetries, it can be carried out just as well for Riesz potentials, the Heisenberg group (or any stratified Lie algebra), and the heat equation (or more general parabolic situations).     

The sum of unitary similarity orbits containing only special operators

Let B(H) be the algebra of bounded linear operator acting on a Hilbert space H (over the complex or real field). Characterization is given to A₁,…,A_k∈B(H) such that for any unitary operators is always in a special class S of operators such as normal operators, self-adjoint operators, unitary operators. As corollaries, characterizations are given to A∈B(H) such that complex, real or nonnegative linear combinations of operators in its unitary orbit U(A)={U^∗AU:Uunitary} always lie in S.     

Tor gives the inverse to the Hilbert function of a graded algebra

The Hilbert generating function of Tor^U(A, A) gives the inverse of the Hilbert generating function of U where U is a graded A algebra.     

Orthogonality of matrices

Let X be a real finite-dimensional normed space with unit sphere S_X and let L(X) be the space of linear operators from X into itself. It is proved that X is an inner product space if and only if for A,C∈L(X)

     

Conditional limiting distribution of Type III elliptical random vectors

In this paper we consider elliptical random vectors in R^d,d≥2 with stochastic representation RAU where R is a positive random radius independent of the random vector U which is uniformly distributed on the unit sphere of R^d and A∈R^d×d is a non-singular matrix. When R has distribution function in the Weibull max-domain of attraction we say that the corresponding elliptical random vector is of Type III. For the bivariate set-up, Berman [Sojurns and Extremes of Stochastic Processes, Wadsworth & Brooks/ Cole, 1992] obtained for Type III elliptical random vectors an interesting asymptotic approximation by conditioning on one component. In this paper we extend Berman's result to Type III elliptical random vectors in R^d. Further, we derive an asymptotic approximation for the conditional distribution of such random vectors.