A Quantitative Version of the Bishop–Phelps Theorem for Operators in Hilbert Spaces

In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 ε 1/2. Then for every bounded linear operator T : H → H and x0 ∈ H with ||T|| = 1 = ||x0|| such that ||Tx0|| 1 ε, there exist xε∈ H and a bounded linear operator S : H → H with||S|| = 1 = ||xε|| such that ||Sxε|| = 1, ||xε-x0|| ≤ (2ε)1/2 + 4(2ε)1/2, ||S-T|| ≤(2ε)1/2.     

Numerical radius and zero pattern of matrices

Let A be an n×n complex matrix and r be the maximum size of its principal submatrices with no off-diagonal zero entries. Suppose A has zero main diagonal and x is a unit n-vector. Then, letting ‖A‖ be the Frobenius norm of A, we show that

|〈Ax,x〉^|2?(1−1/2r−1/2n)‖A^‖2.     

The polynomial-normaloid property for Banach-space operators

An earlier paper (Mh. Math.51, 278–297 (1949)) exploited the property ‖∣Tx‖∣²≤‖∣x‖∣ ‖∣T ² x‖∣, and the same property for polynomials in the operatorT, as an aid in establishing spectral resolutions associated withT. The present paper uses the weaker property ‖∣T‖∣=‖∣T ²‖∣^1/2=..., and its extension to polynomials, for the same purpose. Also considered are the possibility of equivalence between the two types of conditions, and the use of arithmethical hypotheses concerning the eigenvalues.     

Estimates for the Involution of Decomposable Elements of a Complex Banach Algebra

An element a of a complex Banach algebra with unit \(1I\) and with standard conditions on the norm (‖ab‖ ? ‖a‖ · ‖b‖ and ‖\(1I\)‖ = 1) is said to be Hermitian if ‖e ^ita ‖ = 1 for any real number t. An element is said to be decomposable if it admits a representation of the form a + ib in which a and b are Hermitian. The decomposable elements form a Banach Lie algebra (with respect to the commutator). The Hermitian components are determined uniquely, and hence this Lie algebra has the natural involution a + ib = x → x* = a ? ib. One can readily see that ‖x*‖ ? 2‖x‖. Among other things, we prove that ‖ x*‖ ? γ‖x‖, where γ < 2. In fact, the situation is treated in more detail: the original problem is included in a continuous family parametrized by the numerical radius of the element. Finding the exact value of the constant γ is reduced to a variational problem in the theory of entire functions of exponential type. Approximately, γ is equal to 1.92 ± 0.04.     

On the perturbation of the group generalized inverse for a class of bounded operators in Banach spaces

Given a bounded operator A on a Banach space X with Drazin inverse AD and index r, we study the class of group invertible bounded operators B such that I+AD(B−A) is invertible and R(B)∩N(Ar)={0}. We show that they can be written with respect to the decomposition X=R(Ar)⊕N(Ar) as a matrix operator, , where B₁ and are invertible. Several characterizations of the perturbed operators are established, extending matrix results. We analyze the perturbation of the Drazin inverse and we provide explicit upper bounds of ‖B^?−AD‖ and ‖BB^?−ADA‖. We obtain a result on the continuity of the group inverse for operators on Banach spaces.     

A Note on *-paranormal Operators

In this note we show that if either T or T^* is totally *-paranormal then Weyls theorem holds for f(T) for every f , and also a-Weyls theorem holds for f(T) if T is totally *-paranormal. We prove that if either T or T^* is *-paranormal then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.     

Finitely strictly singular operators between James spaces

An operator between Banach spaces is said to be finitely strictly singular if for every ε>0 there exists n such that every subspace E⊆X with dimE?n contains a vector x such that ‖Tx‖<ε‖x‖. We show that, for 1?p<q<∞, the formal inclusion operator from Jp to Jq is finitely strictly singular. As a consequence, we obtain that the strictly singular operator with no invariant subspaces constructed by C. Read is actually finitely strictly singular. These results are deduced from the following fact: if k?n then every k-dimensional subspace of Rn contains a vector x with ‖x?_∞‖=1 such that xmi=i(−1) for some m₁<?<mk.     

Norms and CB norms of Jordan elementary operators

We establish lower bounds for norms and CB-norms of elementary operators on . Our main result concerns the operator T_a,bx=axb+bxa and we show ‖T_a,b‖?‖a‖‖b‖, proving a conjecture of M. Mathieu. We also establish some other results and formulae for ‖T_a,b‖_cb and ‖T_a,b‖ for special cases.     

Some classes of pre-Hilbert algebras with norm-one central idempotent

In this paper we first characterize the pre-Hilbert algebras with a norm-one central idempotent e such that ‖ex‖ = ‖x‖ for any x ∈ A. This generalizes a well-known theorem by Ingelstam asserting that every alternative pre-Hilbert algebra with a unit 1 such that ‖1‖ = 1 is isomorphic to ?, ?, ? or $\mathbb{O}$ . We also show that every power-associative pre-Hilbert algebra satisfying ‖x ²‖ = ‖x‖² for every element has a unique nonzero idempotent, which is a unit element. In fact, the same conclusion will be proved in a more general setting. As application we give some conditions characterizing when a real algebra A, which is a prehilbert space, is isomorphic to one of the Hilbert algebras ?, ?, ? or $\mathbb{O}$ .     

Gaps of operators

We construct examples which distinguish clearly the classes of p-hyponormal operators for 0<p?∞. In addition, we show that those examples classify the classes of w-hyponormal, absolute-p-paranormal, and normaloid operators on the complex Hilbert space. Only a few examples of p-hyponormal operators have been examined. Our technique can provide many examples related to the above operators.     

Local existence of solutions to some degenerate parabolic equation associated with the p-Laplacian

The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: ut(x,t)−Δpu(x,t)−|u|^q−2u(x,t)=f(x,t), (x,t)∈Ω×(0,T), where 2?p<q<+∞, Ω is a bounded domain in RN, is given and Δp denotes the so-called p-Laplacian defined by Δpu:=∇⋅(|∇u|^p−2∇u), with initial data u₀∈Lr(Ω) is proved under r>N(q−p)/p without imposing any smallness on u₀ and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, Lr-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0,T₀] in which the problem admits a solution. More precisely, T₀ depends only on Lr|u₀| and f.     

On the constant and step in Jackson’s inequality for best approximations by trigonometric polynomials and by Haar polynomials

We consider the C*-algebra generated by multidimensional integral operators with (?n)th-order homogeneous kernels and by the operators of multiplication by oscillating coefficients of the form |x|^iα. For this algebra, we construct an operator symbolic calculus and obtain necessary and sufficient conditions for the Fredholm property of an operator in terms of this calculus.     

SVEP and compact perturbations

Let H be a complex separable infinite dimensional Hilbert space. In this paper, we prove that an operator T acting on H is a norm limit of those operators with single-valued extension property (SVEP for short) if and only if T^?, the adjoint of T, is quasitriangular. Moreover, if T^? is quasitriangular, then, given an ε>0, there exists a compact operator K on H with ‖K‖<ε such that T+K has SVEP. Also, we investigate the stability of SVEP under (small) compact perturbations. We characterize those operators for which SVEP is stable under (small) compact perturbations.     

A surjection theorem and a fixed point theorem for a class of positive operators

This paper is concerned with α-convex operators on ordered Banach spaces. A surjection theorem for 1-convex operators in order intervals is established by means of the properties of cone and monotone iterative technique. It is assumed that 1-convex operator A is increasing and satisfies Ay−Ax?M(y−x) for θ?x?y?v₀, where θ denotes the zero element and v₀ is a constant. Moreover, we prove a fixed point theorem for -convex operators by using fixed point theorem of cone expansion. In the end, we apply the fixed point theorem to certain integral equations.     

Boundedness of rough singular integral operators on the Triebel-Lizorkin spaces

We consider the singular integral operator T with kernel K(x)=Ω(x)/n|x| and prove its boundedness on the Triebel-Lizorkin spaces provided that Ω satisfies a size condition which contains the case Ω∈Lr(Sn−1), r>1.     

Irregular vectors of Hilbert space operators

A vector x in a Hilbert space H is called irregular for an operator provided that supn‖Tnx‖=∞ and infn‖Tnx‖=0. We establish some basic properties of operators having irregular vectors and present examples that highlight the relationship, or lack thereof, between irregularity and hypercyclicity.     

Convergent sequences of composition operators

Composition operators Cφ on the Hilbert Hardy space H² over the unit disk are considered. We investigate when convergence of sequences {φn} of symbols, (i.e., of analytic selfmaps of the unit disk) towards a given symbol φ, implies the convergence of the induced composition operators, Cφn→Cφ. If the composition operators Cφn are Hilbert-Schmidt operators, we prove that convergence in the Hilbert-Schmidt norm, ‖Cφn−CφHS_‖→0 takes place if and only if the following conditions are satisfied: ‖φn−φ_‖2→0, ∫1/(1−²|φ|)<∞, and ∫1/(1−²|φn|)→∫1/(1−²|φ|). The convergence of the sequence of powers of a composition operator is studied.     

w* sequential convergence

LetX be an infinite dimensional Banach space, andX* its dual space. Sequences {χ _n ^* } _n=1 ^∞ ?X* which arew* converging to 0 while inf _n ‖x* _n ‖>0, are constructed.     

Operators with rough singular kernels

For α>0, we study the singular integral operators TΩ,α and the Marcinkiewicz integral operator μΩ,α. The kernels of these operators behave like |y|^−n−α near y=0, and contain a distribution Ω on the unit sphere Sn−1. We prove that if Ω∈Hr(Sn−1)(r=(n−1)/(n−1+α)) satisfying certain cancellation condition, then both TΩ,α and μΩ,α can be extend to be the bounded operators from the Sobolev space to the Lebesgue space Lp(Rn). The result improves and extends some known results.