On p-quasi-hyponormal operators

A Hilbert space operator A ∈ B(H) is said to be p-quasi-hyponormal for some 0 < p ? 1, A ∈ p − QH, if A^∗(∣A∣^2p − ∣A^∗∣^2p)A ? 0. If H is infinite dimensional, then operators A ∈ p − QH are not supercyclic. Restricting ourselves to those A ∈ p − QH for which A⁻¹(0) ⊆ A^∗-1(0), A ∈ p_∗ − QH, a necessary and sufficient condition for the adjoint of a pure p_∗ − QH operator to be supercyclic is proved. Operators in p_∗ − QH satisfy Bishop’s property (β). Each A ∈ p_∗ − QH has the finite ascent property and the quasi-nilpotent part H₀(A − λI) of A equals (A − λI)^-1(0) for all complex numbers λ; hence f(A) satisfies Weyl’s theorem, and f(A^∗) satisfies a-Weyl’s theorem, for all non-constant functions f which are analytic on a neighborhood of σ(A). It is proved that a Putnam-Fuglede type commutativity theorem holds for operators in p_∗ − QH.     

A counterexample to a conjecture of Matsaev

We present an effective algorithm for estimating the norm of an operator mapping a low-dimensional ?^p space to a Banach space with an easily computable norm. We use that algorithm to show that Matsaev’s proposed extension of the inequality of John von Neumann is false in case p=4. Matsaev conjectured that for every contraction T on L^p (1<p<∞) one has for any polynomial P

‖P(T)‖_L^p→L^p?‖P(S)‖_{?^p(Z⁺)→?^p(Z⁺)}     

An elementary operator with log-hyponormal, p-hyponormal entries

A Hilbert space operator A∈B(H) is p-hyponormal, A∈(p-H), if |A^∗|^2p?|A|^2p; an invertible operator A∈B(H) is log-hyponormal, A∈(?-H), if log(TT^∗)?log(T^∗T). Let d_AB=δ_AB or ?_AB, where δ_AB∈B(B(H)) is the generalised derivation δ_AB(X)=AX-XB and ?_AB∈B(B(H)) is the elementary operator ?_AB(X)=AXB-X. It is proved that if A,B^∗∈(?-H)∪(p-H), then, for all complex λ, , the ascent of (d_AB-λ)?1, and d_AB satisfies the range-kernel orthogonality inequality ‖X‖?‖X-(d_AB-λ)Y‖ for all X∈(d_AB-λ)^-1(0) and Y∈B(H). Furthermore, isolated points of σ(d_AB) are simple poles of the resolvent of d_AB. A version of the elementary operator E(X)=A₁XA₂-B₁XB₂ and perturbations of d_AB by quasi-nilpotent operators are considered, and Weyl’s theorem is proved for d_AB.     

Paraproducts and Hankel operators of Schatten class via p-John-Nirenberg Theorem

We give an interpolation-free proof of the known fact that a dyadic paraproduct is of Schatten-von Neumann class S_p, if and only if its symbol is in the dyadic Besov space B_p^d. Our main tools are a product formula for paraproducts and a “p-John-Nirenberg-Theorem” due to Rochberg and Semmes.We use the same technique to prove a corresponding result for dyadic paraproducts with operator symbols.Using an averaging technique by Petermichl, we retrieve Peller's characterizations of scalar and vector Hankel operators of Schatten-von Neumann class S_p for 1<p<∞. We then employ vector techniques to characterise little Hankel operators of Schatten-von Neumann class, answering a question by Bonami and Peloso.Furthermore, using a bilinear version of our product formula, we obtain characterizations for boundedness, compactness and Schatten class membership of products of dyadic paraproducts.     

A note on the theorems of M.G. Krein and L.A. Sakhnovich on continuous analogs of orthogonal polynomials on the circle

Continuous analogs of orthogonal polynomials on the circle are solutions of a canonical system of differential equations, introduced and studied by Krein and recently generalized to matrix systems by Sakhnovich. We prove that the continuous analogs of the adjoint polynomials converge in the upper half-plane in the case of L² coefficients, but in general the limit can be defined only up to a constant multiple even when the coefficients are in L^p for any p>2, the spectral measure is absolutely continuous and the Szegö-Kolmogorov-Krein condition is satisfied. Thus, we point out that Krein's and Sakhnovich's papers contain an inaccuracy, which does not undermine known implications from these results.     

Convexity Inequalities for Positive Operators

We prove pointwise convexity (Jensen-type) inequalities of the form Open image in new window where F is a convex function defined on a convex subset of some Banach space X and T is the X-valued extension of a positive operator on some function space. Examples include the pointwise Hölder inequality T(fg) ≤ (Tf ^p )^{1/ p} (Tf ^q )^{1/ q} for a positive sublinear operator T. As applications we consider vector-valued conditional expectation and a ``real'' proof of the Riesz-Thorin theorem for positive operators.     

Convex hypersurfaces and L estimates for Schrödinger equations

This paper is concerned with Schrödinger equations whose principal operators are homogeneous elliptic. When the corresponding level hypersurface is convex, we show the L^p-L^q estimate of the solution operator in the free case. This estimate, combined with the results of fractionally integrated groups, allows us to further obtain the L^p estimate of solutions for the initial data belonging to a dense subset of L^p in the case of integrable potentials.     

Continuity properties for modulation spaces, with applications to pseudo-differential calculus—I

Let M^p,q denote the modulation space with parameters p,q∈[1,∞]. If 1/p₁+1/p₂=1+1/p₀ and 1/q₁+1/q₂=1/q₀, then it is proved that . The result is used to get inclusions between modulation spaces, Besov spaces and Schatten classes in calculus of Ψdo (pseudo-differential operators), and to extend the definition of Toeplitz operators. We also discuss continuity of ambiguity functions and Ψdo in the framework of modulation spaces.     

Approximation methods for common fixed points for a countable family of nonexpansive mappings

Let E=L_p or l_p space, 1<p<∞. Let K be a closed, convex and nonempty subset of E. Let be a family of nonexpansive self-mappings of K. For arbitrary fixed δ∈(0,1), define a family of nonexpansive maps by S_i?(1−δ)I+δT_i where I is the identity map of K. Let . It is proved that the iterative sequence {x_n} defined by: x₀∈K,x_n+1=α_nu+∑_i≥1σ_i,tnS_ix_n,n≥0 converges strongly to a common fixed point of the family where {α_n} and {σ_i,tn} are sequences in (0,1) satisfying appropriate conditions, in each of the following cases: (a) E=l_p,1<p<∞, and (b) E=L_p,1<p<∞ and at least one of the maps T_i’s is demicompact. Our theorems extend the results of [P. Maingé, Approximation methods for common fixed points of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 325 (2007) 469-479] from Hilbert spaces to l_p spaces, 1<p<∞.     

Banach space sequences and projective tensor products

In this paper we use results from the theory of tensor products of Banach spaces to establish the isometry of the space of (1,p)-summing sequences (also known as strongly p-summable sequences) in a Banach space X, the space of nuclear X-valued operators on ?^p and the complete projective tensor product of ?^p with X. Through similar techniques from the theory of tensor products, the isometry of the sequence space L^p〈X〉 (recently introduced in a paper by Bu, Quaestiones Math. (2002), to appear), the space of nuclear X-valued operators on L^p(0,1) (with a suitable equivalent norm) and the complete projective tensor product of L^p(0,1) with X is established. Moreover, we find conditions for the space of (p,q)-summing multipliers to have the GAK-property (generalized AK-property), use multiplier sequences to characterize Banach space valued bounded linear operators on the vector sequence space of absolutely p-summable sequences in a Banach space and present short proofs for results on p-summing multipliers.     

Hypercyclic subspaces and weighted shifts

We first generalize the results of León-Saavedra and Müller (2006) [10] on hypercyclic subspaces to sequences of operators on Fréchet spaces with a continuous norm. Then we study the particular case of iterates of an operator T   and show a simple criterion for having no hypercyclic subspace. Finally we deduce from this criterion a characterization of weighted shifts with hypercyclic subspaces on the spaces l^p$l^p$ or _c0$c_0$, on the space of entire functions and on certain Köthe sequence spaces. We also prove that if P is a non-constant polynomial and D   is the differentiation operator on the space of entire functions then P(D)$P(D)$ possesses a hypercyclic subspace.     

Asymptotic properties of some non-autonomous systems in Banach spaces

Let X be a reflexive Banach space. We introduce the notion of weakly almost nonexpansive sequences (xn)_n?0 in X, and study their asymptotic behavior by showing that the nonempty weak ω-limit set of the sequence (xn/n)_n?1 always lies on a convex subset of a sphere centered at the origin of radius d=limn→∞‖xn/n‖. Subsequently we apply our results to study the asymptotic properties of unbounded trajectories for the quasi-autonomous dissipative system , where A is an accretive (possibly multivalued) operator in X×X, and f−f_∞∈Lp((0,+∞);X) for some f_∞∈X and 1?p<∞. These results extend recent results of J.S. Jung and J.S. Park [J.S. Jung, J.S. Park, Asymptotic behavior of nonexpansive sequences and mean points, Proc. Amer. Math. Soc. 124 (1996) 475-480], and J.S. Jung, J.S. Park, and E.H. Park [J.S. Jung, J.S. Park, E.H. Park, Asymptotic behaviour of generalized almost nonexpansive sequences and applications, Proc. Nonlinear Funct. Anal. 1 (1996) 65-79], as well as our results cited below containing previous results by several authors.     

Catalan and Apéry numbers in residue classes

We estimate character sums with Catalan numbers and middle binomial coefficients modulo a prime p. We use this bound to show that the first at most p^13/2(logp)⁶ elements of each sequence already fall in all residue classes modulo every sufficiently large p, which improves the previously known result requiring p^O(p) elements. We also study, using a different technique, similar questions for sequences satisfying polynomial recurrence relations like the Apéry numbers. We show that such sequences form a finite additive basis modulo p for every sufficiently large prime p.     

The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces

Very few Banach spaces E are known for which the lattice of closed ideals in the Banach algebra of all (bounded, linear) operators on E is fully understood. Indeed, up to now the only such Banach spaces are, up to isomorphism, Hilbert spaces and the sequence spaces c₀ and ?_p for 1?p<∞. We add a new member to this family by showing that there are exactly four closed ideals in for the Banach space E?(⊕?₂ⁿ)_c0, that is, E is the c₀-direct sum of the finite-dimensional Hilbert spaces ?₂¹,?₂²,…,?₂ⁿ,… .     

A monotone iterative technique for stationary and time dependent problems in Banach spaces

An abstract monotone iterative method is developed for operators between partially ordered Banach spaces for the nonlinear problem Lu=Nu and the nonlinear time dependent problem u^′=(L+N)u. Under appropriate assumptions on L and N we obtain maximal and minimal solutions as limits of monotone sequences of solutions of linear problems. The results are illustrated by means of concrete examples.     

A Hardy inequality in the half-space

Here we prove a Hardy-type inequality in the upper half-space which generalize an inequality originally proved by Maz’ya (Sobolev Spaces, Springer, Berlin, 1985, p. 99). Here we present a different proof, which enable us to improve the constant in front of the remainder term. We will also generalize the inequality to the L^p case.     

About computation of the linear complexity of generalized cyclotomic sequences with period p n+1

We propose a computation method for linear complexity of series of generalized cyclotomic sequences with period p ⁿ⁺¹. This method is based on using the polynomial of the classic cyclotomic sequences of period p. We found the linear complexity of generalized cyclotomic sequences corresponding to the classes of biquadratic residues and Hall sequences.     

A spectral sequence for string cohomology

Let X be a 1-connected space with free-loop space ΛX. We introduce two spectral sequences converging towards H^*(ΛX;Z/p) and H^*((ΛX)_hT;Z/p). The E₂-terms are certain non-Abelian-derived functors applied to H^*(X;Z/p). When H^*(X;Z/p) is a polynomial algebra, the spectral sequences collapse for more or less trivial reasons. If X is a sphere it is a surprising fact that the spectral sequences collapse for p=2.     

Poincaré type inequalities on the discrete cube and in the CAR algebra

We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {?1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and Götze. We also get inequalities between the L ^p norms of $ \left\vert \nabla f\right\vert We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {−1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and G?tze. We also get inequalities between the L ^p norms of and moreover L ^p spaces may be replaced by more general ones. Similar results hold true, replacing functions on the cube by matrices in the *-algebra spanned by n fermions and the L ^p norm by the Schatten norm C _p .     

Bases of reproducing kernels in de-Branges spaces

This paper deals with geometric properties of sequences of reproducing kernels related to de-Branges spaces. If b is a nonconstant function in the unit ball of H^∞, and T_b is the Toeplitz operator, with symbol b, then the de-Branges space, H(b), associated to b, is defined by , where H² is the Hardy space of the unit disk. It is equipped with the inner product such that is a partial isometry from H² onto H(b). First, following a work of Ahern-Clark, we study the problem of orthogonal basis of reproducing kernels in H(b). Then we give a criterion for sequences of reproducing kernels which form an unconditional basis in their closed linear span. As far as concerns the problem of complete unconditional basis in H(b), we show that there is a dichotomy between the case where b is an extreme point of the unit ball of H^∞ and the opposite case.