Approximation by Weakly Compact Operators in L1

For every bounded linear operator T in L₁[0, 1] there is an element of best approximation in the ideal of weakly compact operators. We also give some sufficient conditions for ‖T+S‖ = ‖T‖ + ‖S‖, where S and T are L₁-operators.     

A Note on Classes of BANACH Spaces Related to Stable Measures

The problem under consideration is the following: Let S: E′ → L_q, T: E′ → L_p, 0 < q ≦ 2, 0 < p ≦ 2, be operators, ‖Sa‖ ≦ ‖Ta‖, such that, T generates a stable measure on E, i.e., exp (-‖Ta‖ ^p), a ? E′, is the characteristic function of a RADON measure on E. Does this imply, that exp (-‖Sa‖ q), a ? E′, is the characteristic function of a RADON measure, too? In general this is not true provided q or p less than 2. A BANACH space is said to be of (q,p)-cotype if the answer to the above question is “yes”. We establish several properties of this classification and obtain as an application the well-known classes due to MOUCHTARI, TIEN, WERON and MANDREKAR, WERON, Finally we apply our results to so-called S-spaces.     

Fourier type 2 operators with respect to locally compact abelian groups

A linear and bounded operator T between Banach spaces X and Y has Fourier type 2 with respect to a locally compact abelian group G if there exists a constant c > 0 such that∥T ∥₂ ≤ c∥f∥₂ holds for all X‐valued functions f ∈ L^X₂(G) where is the Fourier transform of f. We show that any Fourier type 2 operator with respect to the classical groups has Fourier type 2 with respect to any locally compact abelian group. This generalizes previous special results for the Cantor group and for closed subgroups of ?ⁿ. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Characterization of intermediate values of the triangle inequality II

In [Mineno K., Nakamura Y., Ohwada T., Characterization of the intermediate values of the triangle inequality, Math. Inequal. Appl., 2012, 15(4), 1019–1035] there was established a norm inequality which characterizes all intermediate values of the triangle inequality, i.e. C _n that satisfy 0 ≤ C _n ≤ Σ _j=1 ⁿ ‖x _j ‖ ? ‖Σ _j=1 ⁿ x _j ‖, x ₁,...,x _n ∈ X. Here we study when this norm inequality attains equality in strictly convex Banach spaces.     

The automorphism group of a certain unbounded non-hyperbolic domain

In this paper we determine the automorphism group of the Fock–Bargmann–Hartogs domain _Dn,m$D_{n,m}$ in Cⁿ×C^m${\mathbb{C}}^n\times {\mathbb{C}}^m$ which is defined by the inequality ‖ζ‖²<e^{−μ‖z‖2}${\Vert \zeta \Vert }^2<e^{-\mu {\Vert z\Vert }^2}$.     

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.     

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.     

Compact homomorphisms of regular banach algebras

Let A be a complex commutative Banach algebra and let M_A be the maximal ideal space of A. We say that A has the bounded separating property if there exists a constant C > 0 such that for every two distinct points ?₁, ?₂ ∈ M_A, there is an element a ∈ A for which , and ‖a‖ ? C, where is the Gelfand transform of a ∈ A. We show that if A is a strongly regular Banach algebra with the bounded separating property, then every compact homomorphism from A into another Banach algebra is of finite dimensional range. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

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.     

A lower bound in Nehari’s theorem on the polydisc

By theorems of Ferguson and Lacey (d = 2) and Lacey and Terwilleger (d > 2), Nehari??s theorem (i.e., if H _?? is a bounded Hankel form on H ²(D ^d ) with analytic symbol ??, then there is a function ?? in L ^??(T ^d ) such that ?? is the Riesz projection of g4) is known to hold on the polydisc D ^d for d > 1. A method proposed in Helson??s last paper is used to show that the constant C _d in the estimate ??????_?? ?? C _d ??H _?? ?? grows at least exponentially with d; it follows that there is no analogue of Nehari??s theorem on the infinite-dimensional polydisc.     

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? $W(\mathbb{D})$ .     

Bounds for norms of the matrix inverse and the smallest singular value

In the first part, we obtain two easily calculable lower bounds for ‖A^-1‖, where ‖·‖ is an arbitrary matrix norm, in the case when A is an M-matrix, using first row sums and then column sums. Using those results, we obtain the characterization of M-matrices whose inverses are stochastic matrices. With different approach, we give another easily calculable lower bounds for ‖A^-1‖_∞ and ‖A^-1‖₁ in the case when A is an M-matrix. In the second part, using the results from the first part, we obtain our main result, an easily calculable upper bound for ‖A^-1‖₁ in the case when A is an SDD matrix, thus improving the known bound. All mentioned norm bounds can be used for bounding the smallest singular value of a matrix.     

Interior numerical approximation of boundary value problems with a distributional data

We study the approximation properties of a harmonic function u ∈ H^1?k(Ω), k > 0, on a relatively compact subset A of Ω, using the generalized finite element method (GFEM). If Ω = ??, for a smooth, bounded domain ??, we obtain that the GFEM‐approximation u_S ∈ S of u satisfies ‖u ? u_S‖ ≤ Ch^γ‖u‖, where h is the typical size of the “elements” defining the GFEM‐space S and γ ≥ 0 is such that the local approximation spaces contain all polynomials of degree k + γ. The main technical ingredient is an extension of the classical super‐approximation results of Nitsche and Schatz (Applicable Analysis 2 (1972), 161–168; Math Comput 28 (1974), 937–958). In addition to the usual “energy” Sobolev spaces H¹(??), we need also the duals of the Sobolev spaces H^m(??), m ∈ ?₊. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

For most large underdetermined systems of equations,the minimal 𝓁1‐norm near‐solution approximates the sparsest near‐solution

We consider inexact linear equations y ≈ Φx where y is a given vector in ?ⁿ, Φ is a given n × m matrix, and we wish to find x_0,? as sparse as possible while obeying ‖y ? Φx_0,?‖₂ ≤ ?. In general, this requires combinatorial optimization and so is considered intractable. On the other hand, the ??₁‐minimization problem is convex and is considered tractable. We show that for most Φ, if the optimally sparse approximation x_0,? is sufficiently sparse, then the solution x_1,? of the ??₁‐minimization problem is a good approximation to x_0,?. We suppose that the columns of Φ are normalized to the unit ??₂‐norm, and we place uniform measure on such Φ. We study the underdetermined case where m ～ τn and τ > 1, and prove the existence of ρ = ρ(τ) > 0 and C = C(ρ, τ) so that for large n and for all Φ's except a negligible fraction, the following approximate sparse solution property of Φ holds: for every y having an approximation ‖y ? Φx₀‖₂ ≤ ? by a coefficient vector x₀ ∈ ?^m with fewer than ρ · n nonzeros, This has two implications. First, for most Φ, whenever the combinatorial optimization result x_0,? would be very sparse, x_1,? is a good approximation to x_0,?. Second, suppose we are given noisy data obeying y = Φx₀ + z where the unknown x₀ is known to be sparse and the noise ‖z‖₂ ≤ ?. For most Φ, noise‐tolerant ??₁‐minimization will stably recover x₀ from y in the presence of noise z. We also study the barely determined case m = n and reach parallel conclusions by slightly different arguments. Proof techniques include the use of almost‐spherical sections in Banach space theory and concentration of measure for eigenvalues of random matrices. © 2006 Wiley Periodicals, Inc.     

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.     

A connection between Schur multiplication and Fourier interpolation. II

Given m × n matrices A = [a_jk ] and B = [b_jk ], their Schur product is the m × n matrix A ○ B = [a_jkb_jk ]. For any matrix T, define ‖T‖ _S = max_{X ≠O} ‖T ○ X ‖/‖X ‖ (where ‖·‖ denotes the usual matrix norm). For any complex (2n – 1)‐tuple μ = (μ _{–n +1}, μ _{–n +2}, …, μ _{n –1}), let T_μ be the Hankel matrix [μ –_{n +j +k –1}]_j,k and define ??_μ = {f ∈ L ₁[–π, π] : f? (2j ) = μ_j for –n + 1 ≤ j ≤ n – 1} . It is known that ‖T_μ‖ _S ≤ infequation/tex2gif-inf-18.gif ‖f ‖₁. When equality holds, we say T_μ is distinguished. Suppose now that μ _j ∈ ? for all j and hence that T_μ is hermitian. Then there is a real n × n hermitian unitary X and a real unit vector y such that 〈(T_μ ○ X )y, y 〉 = ‖T_μ ‖_S . We call such a pair a norming pair for T_μ . In this paper, we study norming pairs for real Hankel matrices. Specifically, we characterize the pairs that norm some distinguished Schur multiplier T_μ . We do this by giving necessary and suf.cient conditions for (X, y ) to be a norming pair in the n ‐dimensional case. We then consider the 2‐ and 3‐dimensional cases and obtain further results. These include a new and simpler proof that all real 2 × 2 Hankel matrices are distinguished, and the identi.cation of new classes of 3 × 3 distinguished matrices. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the theory of Besov–Herz spaces and Euler equations

Inspired by Kalton and Wood’s work on group algebras, we describe almost completely contractive algebra homomorphisms from Fourier algebras into Fourier–Stieltjes algebras (endowed with their canonical operator space structure). We also prove that two locally compact groups are isomorphic if and only if there exists an algebra isomorphism T between the associated Fourier algebras (resp. Fourier–Stieltjes algebras) with completely bounded norm \({\left\| T \right\|_{cb}} < \sqrt {3/2} \left( {{\text{resp}}{\text{.}}{{\left\| T \right\|}_{cb}} < \sqrt {5/2} } \right)\). We show similar results involving the norm distortion ‖T‖‖T ^?1‖ with universal but non-explicit bound. Our results subsume Walter’s well-known structural theorems and also Lau’s theorem on the second conjugate of Fourier algebras.     

Saturation on locally compact abelian groups

Let G be an arbitrary locally compact abelian group. It is the purpose of the present paper to establish saturation theorems for approximation processes generated by families (_t)_{t > 0} of complex bounded Radon measures on G and operating on a submodule of the Banach module L^p(G), L^p(G), over the convolution algebra. A basic tool is the Fourier transform method and, in the case p>1 for noncompact G, its interpretation in the context of the theory of quasimeasures on G.     

Distance Functions and Almost Global Solutions of Eikonal Equations

Consider a function u defined on  ⁿ , except, perhaps, on a closed set of potential singularities . Suppose that u solves the eikonal equation ‖Du‖ = 1 in the pointwise sense on  ⁿ \, where Du denotes the gradient of u and ‖·‖ is a norm on  ⁿ with the dual norm ‖·‖_?. For a class of norms which includes the standard p-norms on  ⁿ , 1 < p < ∞, we show that if  has Hausdorff 1-measure zero and n ≥ 2, then u is either affine or a “cone function,” that is, a function of the form u(x) = a ± ‖x ? z‖_?.     

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.