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 $\mathbb{O}$ .     

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.     

A weak-type inequality for uniformly bounded trigonometric polynomials

This paper is devoted to refining the Bernstein inequality. Let D be the differentiation operator. The action of the operator Λ = D/n on the set of trigonometric polynomials T _n is studied: the best constant is sought in the inequality between the measures of the sets {x ∈ T: |Λt(x)| > 1} and {x ∈ T: |t(x)| > 1}. We obtain an upper estimate that is order sharp on the set of uniformly bounded trigonometric polynomials T _n ^C = {t ∈ T _n : ‖t‖ ≤ C}.     

Classification and geometric aspects of vector valued Fourier transforms

It is shown that for any locally compact abelian group ?? and 1 ≤ p ≤ 2, the Fourier type p norm with respect to ?? of a bounded linear operator T between Banach spaces, denoted by ‖T |???^??_p‖, satisfies ‖T |???^??_p‖ ≤ ‖T |???^??_p‖, where ?? is the direct product of ?₂, ?₃, ?₄, … It is also shown that if ?? is not of bounded order then Cⁿ_p ‖T |???^??_p‖ ≤ ‖T |???^??_p‖, where ?? is the circle group, n is a onnegative integer and C^p = . From these inequalities, for any locally compact abelian group ?? ‖T |???^??₂‖ ≤ ‖T |???^??₂‖, and moreover if ?? is not of bounded order then ‖T |???^??₂‖ = ‖T |???^??₂‖. The Hilbertian property and B‐convexity are discussed in the framework of Fourier type p norms. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On summability of weighted Lagrange interpolation. II

The aim of this paper is to continue our investigations started in [15], where we studied the summability of weighted Lagrange interpolation on the roots of orthogonal polynomials with respect to a weight function w. Starting from the Lagrange interpolation polynomials we constructed a wide class of discrete processes which are uniformly convergent in a suitable Banach space (C _ρ, ‖·‖_ρ) of continuous functions (ρ denotes (another) weight). In [15] we formulated several conditions with respect to w, ρ, (C _ρ, ‖·‖_ρ) and to summation methods for which the uniform convergence holds. The goal of this part is to study the special case when w and ρ are Freud-type weights. We shall show that the conditions of results of [15] hold in this case. The order of convergence will also be considered.     

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.     

Finite time extinction of super-Brownian motions with deterministic catalyst

In this paper we consider a super-Brownian motion X with branching mechanism k(x)zα, where k(x) > 0 is a bounded Holder continuous function on Rd and infx∈Rd k(x) = 0. We prove that if k(x) ≥ //x// -l(0 ≤l < ∞) for sufficiently large x, then X has compact support property, and for dimension d = 1, if k(x) ≥exp(-l‖x‖)(0≤l < ∞) for sufficiently large x, then X also has compact support property. The maximal order of k(x) for finite time extinction is different between d = 1, d = 2 and d ≥ 3: it is O(‖x‖-(α+1)) in one dimension, O(‖x‖-2(log‖x‖)-(α+1) ) in two dimensions, and O(‖x‖2) in higher dimensions. These growth orders also turn out to be the maximum order for the nonexistence of a positive solution for 1/2Δu =k(x)uα.     

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.     

Generalized induced norms

Let ‖·‖ be a norm on the algebra ?_n of all n × n matrices over ?. An interesting problem in matrix theory is that “Are there two norms ‖·‖₁ and ‖·‖₂ on ?_n such that ‖A‖ = max|‖Ax‖₂: ‖x‖₁ = 1} for all A ∈ ?_n?” We will investigate this problem and its various aspects and will discuss some conditions under which ‖·‖₁ = ‖·‖₂.     

Small into-isomorphism from L∞ (A, μ) into L∞(B, υ)

In this paper we shall assert that if T is an isomorphism of L_∞(Ω₁, A, μ) into L_∞(Ω₂, B, υ) satisfying the condition ‖T‖·‖T ^?1‖?1+? for ?∈ $\left( {0,\frac{1}{5}} \right)$ , then $\frac{T}{{\parallel T\parallel }}$ is close to an isometry with an error less than 6ε in some conditions.     

Roots of unity and the polynomials with coefficients in{\mathbb{T}}

The maximal order of the coefficients of an arbitrary polynomialP having coefficients in ${\mathbb{T}}$ and the quotient of ‖P‖ _L ⁴ divided by ‖P‖ _L ² depends on the behavior of them in ${\mathbb{T}}$ . In this paper we show thatP can be approximated with another such polynomials which has coefficients as roots of unity.     

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.     

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.     

Three positive solutions of semilinear elliptic equations in exterior cylinder domains

In this paper, assume that h is nonnegative and ‖hL²_‖>0, we prove that if ‖hL²_‖ is sufficiently small, then there are at least three positive solutions of Eq. (1) in an exterior cylinder domain.     

Local rotundity structure of Calderón-Lozanovskii spaces

General results saying that a point x of the unit sphere S(E) of a Köthe space E is an extreme point (a strongly extreme point) [an SU-point] of B(E) if and only if ‖x‖ is an extreme point (a strongly extreme point) [an SU-point] of B(E⁺) and ‖x‖ is a UM-point (a ULUM-point) [nothing more] of E are proved. These results are applied to get criteria for extreme points and SU-points of the unit ball in Caldern-Lozanovski spaces which refer to problem XII from [5]. Strongly extreme points in these spaces are also discussed.     

Note on a conjecture about Bernstein type inequalities for multivariate polynomials

There are fine extensions of the univariate Bernstein-Szeg? inequality for multivariate polynomials considered on a convex domain K. The current one estimates the gradient of the polynomial P at a point x ∈ K by constant times degree, ‖P‖ _C(K) and a geometrical factor. The best constant is within [2, 2√2]. In this note we disprove the conjecture (based on some particular cases) that the best constant is 2.     

Picard dimension of signed radial Kato measures

The Picard dimension dimμ of a signed local Kato measure μ on the punctured unit ball in R^d, d ≥ 2, is the cardinal number of the set of extremal rays of the convex cone of all continuous solutions u ≥ 0 of the time-independent SchrSdinger equation Δu -- uμ = 0 on the punctured ball 0 〈 ｜｜x｜｜ 〈 1, with vanishing boundary values on the sphere ｜｜x｜｜ = 1. Using potential theory associated with the Schrodinger operator we prove, in this paper, that the dimμ for a signed radial Kato measure is 0, 1 or ＋∞. In particular, we obtain the Picard dimension of locally Holder continuous functions P proved by Nakai and Tada by other methods.     

Domination conditions and spectral multiplicity of operators

We study the spectral multiplicity for the direct sum A⊕B of operators A and B on the Banach spaces?X and?Y. Under some domination conditions ‖P(B)‖≦C‖P(A)‖, in particular, ‖B ⁿ ‖≦C‖A ⁿ ‖, n≧0, we prove the addition formulas μ(A⊕B)=μ(A)+μ(B) for spectral multiplicities. We give valuable new applications of the main result of the author’s paper?[12]. We also use the so-called Borel transformation and generalized Duhamel product in calculating the spectral multiplicity of a direct sum of the form T⊕A, where T is a weighted shift operator on the Wiener algebra? $W(\mathbb{D})$ .     

Effective rates of convergence for Lipschitzian pseudocontractive mappings in general Banach spaces

This paper gives an explicit and effective rate of convergence for an asymptotic regularity result ‖Tx_n−x_n‖→0 due to Chidume and Zegeye in 2004 [14] where (x_n) is a certain perturbed Krasnoselski-Mann iteration schema for Lipschitz pseudocontractive self-mappings T of closed and convex subsets of a real Banach space. We also give a qualitative strengthening of the theorem by Chidume and Zegeye, by weakening the assumption of the existence of a fixed point. For the bounded case, our bound is polynomial in the data involved.