Subspace dual super wavelet and Gabor frames

Due to its potential applications in multiplexing techniques, the study of superframes has interested some researchers. This paper addresses dual super wavelet and Gabor frames in the subspace setting. We obtain a basic-equation characterization for subspace dual super wavelet and Gabor frames. In addition, applying this characterization, we derive a procedure that allows for constructing subspace dual super wavelet frames based on certain subspace dual super Gabor frames, and vice versa. Our results are new even in L²(R;C ^L ) setting.     

New characterizations for core inverses in rings with involution

The core inverse for a complex matrix was introduced by O. M. Baksalary and G. Trenkler. D. S. Raki?, N. ?. Din?i? and D. S. Djordjevi? generalized the core inverse of a complex matrix to the case of an element in a ring. They also proved that the core inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible. It is natural to ask when a group invertible element is core invertible. In this paper, we will answer this question. Let R be a ring with involution, we will use three equations to characterize the core inverse of an element. That is, let a, b ∈ R. Then a ∈ R# with a# = b if and only if (ab)* = ab, ba² = a, and ab² = b. Finally, we investigate the additive property of two core invertible elements. Moreover, the formulae of the sum of two core invertible elements are presented.     

The approximation properties determined by operator ideals

We introduce the notion of the right approximation property with respect to an operator ideal A and solve the duality problem for the approximation property with respect to an operator ideal A, that is, a Banach space X has the approximation property with respect to A ^d whenever X* has the right approximation property with respect to an operator ideal A. The notions of the left bounded approximation property and the left weak bounded approximation property for a Banach operator ideal are introduced and new symmetric results are obtained. Finally, the notions of the p-compact sets and the p-approximation property are extended to arbitrary Banach operator ideals. Known results of the approximation property with respect to an operator ideal and the p-approximation property are generalized.     

Verbal width in anabelian groups

The class A of anabelian groups is defined as the collection of finite groups without abelian composition factors. We prove that the commutator word [x₁, x₂] and the power word x₁^p have bounded width in A when p is an odd integer. By contrast, the word x³⁰ does not have bounded width in A. On the other hand, any given word w has bounded width for those groups G ∈ A whose composition factors are sufficiently large as a function of w. In the course of the proof we establish that sufficiently large almost simple groups cannot satisfy w as a coset identity.     

The inclusion of the Schur algebra in <Emphasis Type="Italic">B</Emphasis>(<Emphasis Type="Italic">ℓ</Emphasis><Superscript>2</Superscript>) is not inverse-closed

The Schur algebra is the algebra of operators which are bounded on ? ¹ and on ? ^∞. In this note, we exhibit an element of the group algebra of the free group with two generators, which, as a convolution operator, is invertible in ? ², and whose inverse is not bounded on ? ¹ nor on ? ^∞. In particular, this shows that the Schur algebra is not inverse-closed.     

The Corona problem on a complete ultrametric algebraically closed field

Let IK be a complete ultrametric algebraically closed field and let A be the Banach IK-algebra of bounded analytic functions in the ”open” unit disk D of IK provided with the Gauss norm. Let Mult(A, ‖. ‖) be the set of continuous multiplicative semi-norms of A provided with the topology of simple convergence, let Mult _m (A, ‖. ‖) be the subset of the ? ∈ Mult(A, ‖. ‖) whose kernel is amaximal ideal and let Mult ₁(A, ‖. ‖) be the subset of the ? ∈ Mult(A, ‖. ‖) whose kernel is a maximal ideal of the form (x ? a)A with a ∈ D. By analogy with the Archimedean context, one usually calls ultrametric Corona problem the question whether Mult ₁(A, ‖. ‖) is dense in Mult _m (A, ‖. ‖). In a previous paper, it was proved that when IK is spherically complete, the answer is yes. Here we generalize this result to any algebraically closed complete ultrametric field, which particularly applies to ? _p . On the other hand, we also show that the continuous multiplicative seminorms whose kernel are neither a maximal ideal nor the zero ideal, found by Jesus Araujo, also lie in the closure of Mult ₁(A, ‖. ‖), which suggest that Mult ₁(A, ‖. ‖)might be dense in Mult(A, ‖. ‖).     

Certain Subsets on Which Every Bounded Convex Function Is Continuous

To guarantee every real-valued convex function bounded above on a set is continuous, how ＂thick＂ should the set be？ For a symmetric set A in a Banach space E,the answer of this paper is： Every real-valued convex function bounded above on A is continuous on E if and only if the following two conditions hold： i） spanA has finite co-dimentions and ii） coA has nonempty relative interior. This paper also shows that a subset A C E satisfying every real-valued convex function bounded above on A is continuous on E if （and only if） every real-valued linear functional bounded above on A is continuous on E, which is also equivalent to that every real-valued convex function bounded on A is continuous on E.     

On some properties of continuous <Emphasis Type="Italic">g</Emphasis>-frames and Riesz-type continuous <Emphasis Type="Italic">g</Emphasis>-frames

In this paper we provide some necessary and sufficient conditions under which, a family of bounded operators is a continuous g-frame (Riesz-type continuous g-frame). Also, we study stability of duals of continuous g-frames.     

Symmetrization of Cuntz’ Picture for the Kasparov <Emphasis Type="Italic">KK</Emphasis>-Bifunctor

Given C^*-algebras A and B, we generalize the notion of a quasi-homomorphism from A to B in the sense of Cuntz by considering quasi-homomorphisms from some C^*-algebra C to B such that C surjects onto A and the two maps forming the quasi-homomorphism agree on the kernel of this surjection. Under an additional assumption, the group of homotopy classes of such generalized quasi-homomorphisms coincides with KK(A, B). This makes the definition of the Kasparov bifunctor slightly more symmetric and provides more flexibility in constructing elements of KK-groups. These generalized quasi-homomorphisms can be viewed as pairs of maps directly from A (instead of various C’s), but these maps need not be *-homomorphisms.     

A note on generalized Lie derivations of prime rings

Let R be a prime ring of characteristic not 2, A be an additive subgroup of R, and F, T, D, K: A-R be additive maps such that F([x, y]) = F(x)_y-yK(x)-T(y)_{x + x}D(y) for all x, yEA. Our aim is to deal with this functional identity when A is R itself or a noncentral Lie ideal of R. Eventually, we are able to describe the forms of the mappings F, T, D, and K in case A = R with deg(R) > 3 and also in the case A is a noncentral Lie ideal and deg(R) > 9. These enable us in return to characterize the forms of both generalized Lie derivations, D-Lie derivations and Lie centralizers of R under some mild assumptions. Finally, we give a generalization of Lie homomorphisms on Lie ideals.     

On homogenization for non-self-adjoint locally periodic elliptic operators

In this note we consider the homogenization problem for a matrix locally periodic elliptic operator on R ^d of the form A ^ε = ?divA(x, x/ε)?. The function A is assumed to be Hölder continuous with exponent s ∈ [0, 1] in the “slow” variable and bounded in the “fast” variable. We construct approximations for (A ^ε ? μ)^?1, including one with a corrector, and for (?Δ) ^s/2(A ^ε ? μ)^?1 in the operator norm on L ₂(R ^d ) ⁿ . For s ≠ 0, we also give estimates of the rates of approximation.     

Right Markov Processes and Systems of Semilinear Equations with Measure Data

In the paper we prove the existence of probabilistic solutions to systems of the form ?Au = F(x, u) + μ, where F satisfies a generalized sign condition and μ is a smooth measure. As for A we assume that it is a generator of a Markov semigroup determined by a right Markov process whose resolvent is order compact on L¹. This class includes local and nonlocal operators corresponding to Dirichlet forms as well as some operators which are not in the variational form. To study the problem we introduce new concept of compactness property relating the underlying Markov process to almost everywhere convergence. We prove some useful properties of the compactness property and provide its characterization in terms of Meyer’s property (L) of Markov processes and in terms of order compactness of the associated resolvent.     

A generalization of sumset and its applications

Let A be a nonempty finite subset of an additive abelian group G and let r and h be positive integers. The generalized h-fold sumset of A, denoted by \(h^{(r)}A\), is the set of all sums of h elements of A, where each element appears in a sum at most r times. The direct problem for \(h^{(r)}A\) is to find a lower bound for \(|h^{(r)}A|\) in terms of |A|. The inverse problem for \(h^{(r)}A\) is to determine the structure of the finite set A for which \(|h^{(r)}A|\) is minimal with respect to some fixed value of |A|. If \(G = \mathbb {Z}\), the direct and inverse problems are well studied. In case of \(G = \mathbb {Z}/p\mathbb {Z}\), p a prime, the direct problem has been studied very recently by Monopoli (J. Number Theory, 157 (2015) 271–279). In this paper, we express the generalized sumset \(h^{(r)}A\) in terms of the regular and restricted sumsets. As an application of this result, we give a new proof of the theorem of Monopoli and as the second application, we present new proofs of direct and inverse theorems for the case \(G = \mathbb {Z}\).     

Perturbations of strongly continuous operator semigroups,and matrix Muckenhoupt weights

Let A and A ₀ be linear continuously invertible operators on a Hilbert space ? such that A ^?1 ? A ₀ ^?1 has finite rank. Assuming that σ(A ₀) = ? and that the operator semigroup V ₊(t) = exp{iA ₀ t}, t ≥ 0, is of class C ₀, we state criteria under which the semigroups U _±(t) = exp{±iAt}, t ≥ 0, are of class C ₀ as well. The analysis in the paper is based on functional models for nonself-adjoint operators and techniques of matrix Muckenhoupt weights.     

Differences of Idempotents In <Emphasis Type="Italic">C</Emphasis>*-Algebras and the Quantum Hall Effect

Let ? be a trace on the unital C*-algebra A and M _? be the ideal of the definition of the trace ?. We obtain a C*analogue of the quantum Hall effect: if P,Q ∈ A are idempotents and P ? Q ∈ M _? , then ?((P ? Q)²ⁿ⁺¹) = ?(P ? Q) ∈ R for all n ∈ N. Let the isometries U ∈ A and A = A*∈ A be such that I+A is invertible and U-A ∈ M _? with ?(U-A) ∈ R. Then I-A, I?U ∈ M _? and ?(I?U) ∈ R. Let n ∈ N, dimH = 2n + 1, the symmetry operators U, V ∈ B(H), and W = U ? V. Then the operator W is not a symmetry, and if V = V*, then the operator W is nonunitary.     

On a Fitting length conjecture without the coprimeness condition

Let A be a finite nilpotent group acting fixed point freely by automorphisms on the finite solvable group G. It is conjectured that the Fitting length of G is bounded by the number of primes dividing the order of A, counted with multiplicities. The main result of this paper shows that the conjecture is true in the case where A is cyclic of order p ⁿ q, for prime numbers p and q coprime to 6 and G has abelian Sylow 2-subgroups.     

Complete bounded λ-hypersurfaces in the weighted volume-preserving mean curvature flow

In this paper, we study the complete bounded λ-hypersurfaces in the weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded λ-hypersurfaces with |A|≤α and get some applications of the volume comparison theorem. Secondly, we consider the relation among λ, extrinsic radius k, intrinsic diameter d, and dimension n of the complete λ-hypersurface,and we obtain some estimates for the intrinsic diameter and the extrinsic radius. At last, we get some topological properties of the bounded λ-hypersurface with some natural and general restrictions.     

Nonsingularity,positive definiteness,and positive invertibility under fixed-point data rounding

For a real square matrix A and an integer d ? 0, let A _(d) denote the matrix formed from A by rounding off all its coefficients to d decimal places. The main problem handled in this paper is the following: assuming that A _(d) has some property, under what additional condition(s) can we be sure that the original matrix A possesses the same property? Three properties are investigated: nonsingularity, positive definiteness, and positive invertibility. In all three cases it is shown that there exists a real number α(d), computed solely from A _(d) (not from A), such that the following alternative holdsif d > α(d), then nonsingularity (positive definiteness, positive invertibility) of A _(d) implies the same property for A if d < α(d) and A _(d) is nonsingular (positive definite, positive invertible), then there exists a matrix A′ with A′_(d) = A _(d) which does not have the respective property.For nonsingularity and positive definiteness the formula for α(d) is the same and involves computation of the NP-hard norm ‖ · ‖_∞,1; for positive invertibility α(d) is given by an easily computable formula.     

Globally convergent Jacobi methods for positive definite matrix pairs

The paper derives and investigates the Jacobi methods for the generalized eigenvalue problem A x = λ B x, where A is a symmetric and B is a symmetric positive definite matrix. The methods first “normalize” B to have the unit diagonal and then maintain that property during the iterative process. The global convergence is proved for all such methods. That result is obtained for the large class of generalized serial strategies from Hari and Begovi? Kova? (Trans. Numer. Anal. (ETNA) 47, 107–147, 2017). Preliminary numerical tests confirm a high relative accuracy of some of those methods, provided that both matrices are positive definite and the spectral condition numbers of Δ _A AΔ _A and Δ _B BΔ _B are small, for some nonsingular diagonal matrices Δ _A and Δ _B .     

Matrix semigroups whose ring commutators have real spectra are realizable

We introduce the numerical spectrum \(\sigma _n(A)\subseteq {\mathbb {C}}\) of an (unbounded) linear operator A on a Banach space X and study its properties. Our definition is closely related to the numerical range W(A) of A and always yields a superset of W(A). In the case of bounded operators on Hilbert spaces, the two notions coincide. However, unlike the numerical range, \(\sigma _n(A)\) is always closed, convex and contains the spectrum of A. In the paper we strongly emphasise the connection of our approach to the theory of \(C_0\)-semigroups.