Contractions Satisfying the Absolute Value Property

Let B(H) denote the algebra of operators on a complex Hilbert space H, and let U denote the class of operators which satisfy the absolute value condition . It is proved that if is a contraction, then either A has a nontrivial invariant subspace or A is a proper contraction and the nonnegative operator is strongly stable. A Putnam-Fuglede type commutativity theorem is proved for contractions A in , and it is shown that if normal subspaces of . It is proved that if are reducing, then every compact operator in the intersection of the weak closure of the range of the derivation with the commutant of A* is quasinilpotent.     

The general dévissage theorem for Witt groups of schemes

Let be a closed subscheme of the noetherian scheme X. We show that if X has a dualizing complex then there exists a dualizing complex of Z such that there is an isomorphism of coherent Witt groups for all . Received: 3 March 2006     

On the Range of the Aluthge Transform

Let be the algebra of all bounded linear operators on a complex separable Hilbert space For an operator let be the Aluthge transform of T and we define for all where T = U|T| is a polar decomposition of T. In this short note, we consider an elementary property of the range of Δ. We prove that R(Δ) is neither closed nor dense in However R(Δ) is strongly dense if is infinite dimensional. An erratum to this article is available at .     

<Emphasis Type="Italic">p</Emphasis>-Laplace Operator and Diameter of Manifolds

Let be a compact Riemannian manifold without boundary. In this paper, we consider the first nonzero eigenvalue of the p-Laplacian and we prove that the limit of when is 2/d(M), where d(M) is the diameter of M. Moreover, if is an oriented compact hypersurface of the Euclidean space or , we prove an upper bound of in terms of the largest principal curvature κ over M. As applications of these results, we obtain optimal lower bounds of d(M) in terms of the curvature. In particular, we prove that if M is a hypersurface of then: . Mathematics Subject Classifications (2000): 53A07, 53C21.     

Linear Maps Preserving Generalized Invertibility

Let H be an infinite-dimensional complex separable Hilbert space and the algebra of all bounded linear operators on H. Let be a bijective continuous unital linear map preserving generalized invertibility in both directions. Then the ideal of all compact operators is invariant under ϕ and the induced linear map on the Calkin algebra is either an automorphism or an antiautomorphism.     

Dicritical holomorphic flows on Stein manifolds

We study holomorphic flows on Stein manifolds. We prove that a holomorphic flow with isolated singularities and a dicritical singularity of the form on a Stein manifold with , is globally analytically linearizable; in particular M is biholomorphic to . A complete stability result for periodic orbits is also obtained. Bruno Scárdua: Partially supported by ICTP-Trieste-Italy. Received: 27 September 2006     

Additive Maps Preserving Local Spectrum

Let X be a complex Banach space, and let be the space of bounded operators on X. Given and x ∈ X, denote by σ_T (x) the local spectrum of T at x. We prove that if is an additive map such that

Grothendieck’s comparison theorem and multivariable Fredholm theory

We use a variant of Grothendieck’s comparison theorem to show that, for a Fredholm tuple T ∈ L(X)ⁿ on a complex Banach space, there are isomorphisms . We conclude that a Fredholm tuple T ∈ L(X)ⁿ satisfies Bishop’s property (β) at z = 0 if and only if the vanishing conditions hold for . We apply these observations and results from commutative algebra to show that a graded tuple on a Hilbert space is Fredholm if and only if it satisfies Bishop’s property (β) at z = 0 and that, in this case, its cohomology groups can grow at most like k^p. Received: 14 January 2009     

Operator Matrices: SVEP and Weyl’s Theorem

It is known that if and are Banach space operators with the single-valued extension property, SVEP, then the matrix operator has SVEP for every operator and hence obeys Browder’s theorem. This paper considers conditions on operators A, B, and M₀ ensuring Weyls theorem for operators M_C.     

Some Properties of Essential Spectra of a Positive Operator

Let E be a Banach lattice, T be a bounded operator on E. The Weyl essential spectrum σ_ew(T) of the operator T is a set , where is a set of all compact operators on E. In particular for a positive operator T next subsets of the spectrum

Lie and Jordan Ideals in Reflexive Algebras

A CDCSL algebra is a reflexive operator algebra with completely distributive and commutative subspace lattice. In this paper, we show, for a weakly closed linear subspace of a CDCSL algebra , that is a Lie ideal if and only if for all invertibles A in , and that is a Jordan ideal if and only if it is an associative ideal.     

Common periodic trajectories of interval maps

We prove that for any continuous piecewise monotone or smooth interval map f and any subset of the set of periods of periodic trajectories of f, there is another map such that the set of periods of periodic trajectories common for f and , which is denoted by , coincides with . At the same time, for each integer , there exists a continuous map f such that for any map if is an infinite set. Dedicated to Vladimir Igorevich Arnold     

C*-Algebras of Integral Operators with Piecewise Slowly Oscillating Coefficients and Shifts Acting Freely

We establish a symbol calculus for the C*-subalgebra of generated by the operators of multiplication by slowly oscillating and piecewise continuous functions and the operators where is the Cauchy singular integral operator and The C*-algebra is invariant under the transformations

Normal growth of large groups

For a finitely generated group $\Gamma$, denote by the number of normal subgroups of index n. A. Lubotzky proved that for the free group F_r of rank r, is of type n^logn. We show that the same is true for a much larger class of groups. On the other hand we show that for almost all n, the inequality < holds true for every r-generated group $\Gamma$.Received: 30 October 2002     

Solution of the Truncated Hyperbolic Moment Problem

Let Q(x, y) = 0 be an hyperbola in the plane. Given real numbers β ≡ β ⁽²ⁿ⁾={ β _ij } _{i,j ≥ 0,i+j ≤ 2n} , with β₀₀ > 0, the truncated Q-hyperbolic moment problem for β entails finding necessary and sufficient conditions for the existence of a positive Borel measure μ, supported in Q(x, y) = 0, such that We prove that β admits a Q-representing measure μ (as above) if and only if the associated moment matrix is positive semidefinite, recursively generated, has a column relation Q(X,Y) = 0, and the algebraic variety associated to β satisfies card In this case, if then β admits a rank -atomic (minimal) Q-representing measure; if then β admits a Q-representing measure μ satisfying      

Extreme values of a portfolio of Gaussian processes and a trend

We consider the extreme values of a portfolio of independent continuous Gaussian processes ( ) which are asymptotically locally stationary, with expectations and variances , and a trend for some constants with . We derive the probability for , which may be interpreted as ruin probability. AMS 2000 Subject Classifications Primary—60G15, 62G32, 91B28     

Complete Positivity of Elementary Operators on C*-Algebras

Let be a C*-algebra. We obtain some conditions that are equivalent to the statement that every n-positive elementary operator on is completely positive.     

Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Boundedness of Marcinkiewicz Integral

A weighted norm inequality for the Marcinkiewicz integral operator is proved when belongs to . We also give the weighted L^p-boundedness for a class of Marcinkiewicz integral operators with rough kernels and related to the Littlewood-Paley -function and the area integral S, respectively.     

Parametrization of Contractive Block Operator Matrices and Passive Discrete-Time Systems

Passive linear systems τ = have their transfer function in the Schur class S . Using a parametrization of contractive block operators the transfer function is connected to the Sz.-Nagy–Foiaş characteristic function of the contraction A. This gives a new aspect and some explicit formulas for studying the interplay between the system τ and the functions and . The method leads to some new results for linear passive discrete-time systems. Also new proofs for some known facts in the theory of these systems are obtained. Dedicated to Eduard Tsekanovskiĭ on the occasion of his seventieth birthday This work was supported by the Research Institute for Technology at the University of Vaasa. The first author was also supported by the Academy of Finland (projects 212146, 117617) and the Dutch Organization for Scientific Research N.W.O. (B 61-553). Received: December 22, 2006. Revised: February 6, 2007.     

Equivalence Bimodule between Spherical Noncommutative Tori

Let n _i ,m _j > 1. In [5], the sperical noncommutative torus was defined by twisting in by a totally skew multiplier p on for T ^pd a pd-homogeneous C*-algebra over . It is shown that is strongly Morita equivalent to . This work is supported by Grant No. 1999-2-102-001-3 from the interdisciplinary research program year of the KOSEF