Bochner representable operators on Banach function spaces

Let \((E,\Vert \cdot \Vert _E)\) be a Banach function space, \(E'\) the Köthe dual of E and \((X,\Vert \cdot \Vert _X)\) be a Banach space. It is shown that every Bochner representable operator \(T:E\rightarrow X\) maps relatively \(\sigma (E,E')\)-compact sets in E onto relatively norm compact sets in X. If, in particular, the associated norm \(\Vert \cdot \Vert _{E'}\) on \(E'\) is order continuous, then every Bochner representable operator \(T:E\rightarrow X\) is \((\gamma _E,\Vert \cdot \Vert _X)\)-compact, where \(\gamma _E\) stands for the natural mixed topology on E. Applications to Bochner representable operators on Orlicz spaces are given.     

Orthogonal-gradings on $$H^*$$-algebras

We study set-gradings on proper \(H^*\)-algebras A, which are compatible with the involution and the inner product of A, that will be called orthogonal-gradings. If A is an arbitrary \(H^*\)-algebra with a fine grading, we obtain a (fine) orthogonal-graded version of the main structure theorem for proper arbitrary \(H^*\)-algebras. If A is associative, we show that any fine orthogonal-grading is either a group-grading or a (non-group grading) Peirce decomposition of A respect to a family of orthogonal projections. If A is alternative, we prove that any fine orthogonal-grading is either a fine orthogonal-grading of a (proper) associative \(H^*\)-algebra, or a \({\mathbb Z}_2^3\)-grading of the complex octonions \({\mathbb O}\) or a non-group grading which is a refinement of the Peirce decomposition of \({\mathbb O}\) respect to its family of orthogonal projections. Finally, we also show that any orthogonal-grading on the real octonion division algebra is necessarily a group-grading.     

Reconstruction Algorithms for a Class of Restricted Ray Transforms Without Added Singularities

Let X and \(X^*\) denote a restricted ray transform along curves and a corresponding backprojection operator, respectively. Theoretical analysis of reconstruction from the data Xf is usually based on a study of the composition \(X^* D X\), where D is some local operator (usually a derivative). If \(X^*\) is chosen appropriately, then \(X^* D X\) is a Fourier integral operator (FIO) with singular symbol. The singularity of the symbol leads to the appearance of artifacts (added singularities) that can be as strong as the original (or, useful) singularities. By choosing D in a special way one can reduce the strength of added singularities, but it is impossible to get rid of them completely. In the paper we follow a similar approach, but make two changes. First, we replace D with a nonlocal operator \(\tilde{D}\) that integrates Xf along a curve in the data space. The result \(\tilde{D} Xf\) resembles the generalized Radon transform R of f. The function \(\tilde{D} Xf\) is defined on pairs \((x_0,\Theta )\in U\times S^2\), where \(U\subset {\mathbb R}^3\) is an open set containing the support of f, and \(S^2\) is the unit sphere in \({\mathbb R}^3\). Second, we replace \(X^*\) with a backprojection operator \(R^*\) that integrates with respect to \(\Theta \) over \(S^2\). It turns out that if \(\tilde{D}\) and \(R^*\) are appropriately selected, then the composition \(R^* \tilde{D} X\) is an elliptic pseudodifferential operator of order zero with principal symbol 1. Thus, we obtain an approximate reconstruction formula that recovers all the singularities correctly and does not produce artifacts. The advantage of our approach is that by inserting \(\tilde{D}\) we get access to the frequency variable \(\Theta \). In particular, we can incorporate suitable cut-offs in \(R^*\) to eliminate bad directions \(\Theta \), which lead to added singularities.     

Persistence of Banach lattices under nonlinear order isomorphisms

Ordered vector spaces E and F are said to be order isomorphic if there is a (not necessarily linear) bijection \(T: E\rightarrow F\) such that \(x\ge y\) if and only if \(Tx \ge Ty\) for all \(x,y \in E\). We investigate some situations under which an order isomorphism between two Banach lattices implies the persistence of some linear lattice structure. For instance, it is shown that if a Banach lattice E is order isomorphic to C(K) for some compact Hausdorff space K, then E is (linearly) isomorphic to C(K) as a Banach lattice. Similar results hold for Banach lattices order isomorphic to \(c_{0}\), and for Banach lattices that contain a closed sublattice order isomorphic to \(c_{0}\).     

Popular edges and dominant matchings

Given a bipartite graph \(G = (A \cup B,E)\) with strict preference lists and given an edge \(e^* \in E\), we ask if there exists a popular matching in G that contains \(e^*\). We call this the popular edge problem. A matching M is popular if there is no matching \(M'\) such that the vertices that prefer \(M'\) to M outnumber those that prefer M to \(M'\). It is known that every stable matching is popular; however G may have no stable matching with the edge \(e^*\). In this paper we identify another natural subclass of popular matchings called “dominant matchings” and show that if there is a popular matching that contains the edge \(e^*\), then there is either a stable matching that contains \(e^*\) or a dominant matching that contains \(e^*\). This allows us to design a linear time algorithm for identifying the set of popular edges. When preference lists are complete, we show an \(O(n^3)\) algorithm to find a popular matching containing a given set of edges or report that none exists, where \(n = |A| + |B|\).     

Generalized derivations on some convolution algebras

Let G be a locally compact abelian group, \(\omega \) be a weighted function on \({\mathbb {R}}^+\), and let \(\mathfrak {D}\) be the Banach algebra \(L_0^\infty (G)^*\) or \(L_0^\infty (\omega )^*\). In this paper, we investigate generalized derivations on the noncommutative Banach algebra \(\mathfrak {D}\). We characterize \(\textsf {k}\)-(skew) centralizing generalized derivations of \(\mathfrak {D}\) and show that the zero map is the only \(\textsf {k}\)-skew commuting generalized derivation of \(\mathfrak {D}\). We also investigate the Singer–Wermer conjecture for generalized derivations of \(\mathfrak {D}\) and prove that the Singer–Wermer conjecture holds for a generalized derivation of \(\mathfrak {D}\) if and only if it is a derivation; or equivalently, it is nilpotent. Finally, we investigate the orthogonality of generalized derivations of \(L_0^\infty (\omega )^*\) and give several necessary and sufficient conditions for orthogonal generalized derivations of \(L_0^\infty (\omega )^*\).     

Simultaneous power factorization in modules over Banach algebras

Let A be a Banach algebra with a bounded left approximate identity \(\{e_\lambda \}_{\lambda \in \Lambda }\), let \(\pi \) be a continuous representation of A on a Banach space X, and let S be a non-empty subset of X such that \(\lim _{\lambda }\pi (e_\lambda )s=s\) uniformly on S. If S is bounded, or if \(\{e_\lambda \}_{\lambda \in \Lambda }\) is commutative, then we show that there exist \(a\in A\) and maps \(x_n: S\rightarrow X\) for \(n\ge 1\) such that \(s=\pi (a^n)x_n(s)\) for all \(n\ge 1\) and \(s\in S\). The properties of \(a\in A\) and the maps \(x_n\), as produced by the constructive proof, are studied in some detail. The results generalize previous simultaneous factorization theorems as well as Allan and Sinclair’s power factorization theorem. In an ordered context, we also consider the existence of a positive factorization for a subset of the positive cone of an ordered Banach space that is a positive module over an ordered Banach algebra with a positive bounded left approximate identity. Such factorizations are not always possible. In certain cases, including those for positive modules over ordered Banach algebras of bounded functions, such positive factorizations exist, but the general picture is still unclear. Furthermore, simultaneous pointwise power factorizations for sets of bounded maps with values in a Banach module (such as sets of bounded convergent nets) are obtained. A worked example for the left regular representation of \(\mathrm {C}_0({\mathbb R})\) and unbounded S is included.     

<Emphasis Type="Italic">q</Emphasis>-Frequent hypercyclicity in spaces of operators

We provide conditions for a linear map of the form \(C_{R,T}(S)=RST\) to be q-frequently hypercyclic on algebras of operators on separable Banach spaces. In particular, if R is a bounded operator satisfying the q-frequent hypercyclicity criterion, then the map \(C_{R}(S)=RSR^*\) is shown to be q-frequently hypercyclic on the space \(\mathcal {K}(H)\) of all compact operators and the real topological vector space \(\mathcal {S}(H)\) of all self-adjoint operators on a separable Hilbert space H. Further we provide a condition for \(C_{R,T}\) to be q-frequently hypercyclic on the Schatten von Neumann classes \(S_p(H)\). We also characterize frequent hypercyclicity of \(C_{M^*_\varphi ,M_\psi }\) on the trace-class of the Hardy space, where the symbol \(M_\varphi \) denotes the multiplication operator associated to \(\varphi \).     

The dual Radon–Nikodym property for finitely generated Banach <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>)-modules

We extend the well-known criterion of Lotz for the dual Radon–Nikodym property (RNP) of Banach lattices to finitely generated Banach C(K)-modules and Banach C(K)-modules of finite multiplicity. Namely, we prove that if X is a Banach space from one of these classes then its Banach dual \(X^\star \) has the RNP iff X does not contain a closed subspace isomorphic to \(\ell ^1\).     

Levi Extension Theorems for Meromorphic Functions of Weak Type in Infinite Dimension

The aim of paper is to give some results, that prepare for studying the problem on cross theorems for separately \((\cdot , W)\)-meromorphic functions. Some general versions of extension theorem of Levi type are extended to the classes of meromorphic functions f on \(D \times (\Delta _r {\setminus } \overline{\Delta })\) with values in a locally convex space F. Here, the function f is assumed that, for each \(z \in D_*,\) the function \(f_z = f(z, \cdot )\) has a (F, W)-meromorphic extension to \(\Delta _r,\) where F is either a locally (or sequentially) complete locally convex space or a Fréchet space, the space \(W \subseteq F'\) is separating (or determines boundedness), \(\Delta _r = \{\lambda \in {\mathbb C}: |\lambda | < r\}\) with \(r > 1, \Delta = \Delta _1\) and D is either a domain in \({\mathbb C}^n\) or a balanced domain in a Fréchet space containing a non-pluripolar balanced convex compact subset, \(D_*\) is dense in D.     

Approximation of entire functions of exponential type by exponential sums<Superscript>*</Superscript>

Let K be a compact set in \( {{\mathbb R}^n} \). For \( 1 \leqslant p \leqslant \infty \), the Bernstein space \( B_K^p \) is the Banach space of all functions \( f \in {L^p}\left( {{{\mathbb R}^n}} \right) \)such that their Fourier transform in a distributional sense is supported on K. If \( f \in B_K^p \), then f is continuous on \( {{\mathbb R}^n} \) and has an extension onto the complex space \( {{\mathbb C}^n} \) to an entire function of exponential type K. We study the approximation of functions in \( B_K^p \) by finite τ -periodic exponential sums of the form

$ \sum\limits_m {{c_m}{e^{2\pi {\text{i}}\left( {x,m} \right)/\tau }}} $

in the \( {L^p}\left( {\tau {{\left[ { - 1/2,1/2} \right]}^n}} \right) \)-norm as τ → ∞ when K is a polytope in \( {{\mathbb R}^n} \).

     

A concave–convex problem with a variable operator

We study the following elliptic problem \(-A(u) = \lambda u^q\) with Dirichlet boundary conditions, where \(A(u) (x) = \Delta u (x) \chi _{D_1} (x)+ \Delta _p u(x) \chi _{D_2}(x)\) is the Laplacian in one part of the domain, \(D_1\), and the p-Laplacian (with \(p>2\)) in the rest of the domain, \(D_2 \). We show that this problem exhibits a concave–convex nature for \(1<q<p-1\). In fact, we prove that there exists a positive value \(\lambda ^*\) such that the problem has no positive solution for \(\lambda > \lambda ^*\) and a minimal positive solution for \(0<\lambda < \lambda ^*\). If in addition we assume that p is subcritical, that is, \(p<2N/(N-2)\) then there are at least two positive solutions for almost every \(0<\lambda < \lambda ^*\), the first one (that exists for all \(0<\lambda < \lambda ^*\)) is obtained minimizing a suitable functional and the second one (that is proven to exist for almost every \(0<\lambda < \lambda ^*\)) comes from an appropriate (and delicate) mountain pass argument.     

Closed range and nonclosed range adjointable operators on Hilbert $$C^*$$-modules

In this paper, we show that for a positive operator A on a Hilbert \(C^*\)-module \( \mathscr {E} \), the range \( \mathscr {R}(A) \) of A is closed if and only if \( \mathscr {R}(A^\alpha ) \) is closed for all \(\alpha \in (0,1)\cup (1,+\,\infty )\), and this occurs if and only if \( \mathscr {R}(A)=\mathscr {R}(A^\alpha ) \) for all \(\alpha \in (0,1)\cup (1,+\,\infty )\). As an application, we prove that for an adjontable operator A if \(\mathscr {R}(A)\) is nonclosed, then \(\dim \left( \overline{\mathscr {R}(A)}/\mathscr {R}(A)\right) =+\,\infty \). Finally, we show that for an adjointable operator A if \( \overline{\mathscr {R}(A^*) } \) is orthogonally complemented in \( \mathscr {E} \), then under certain coditions there exists an idempotent C and a unique operator X such that \( XAX=X, AXA=CA, AX=C \) and \( XA=P_{A^*} \), where \( P_{A^*} \) is the orthogonal projection of \( \mathscr {E} \) onto \( \overline{\mathscr {R}(A^*)}\).     

Best proximity point theorems in the frameworks of fairly and proximally complete spaces

Let us contemplate the problem of solving the linear or non-linear equations of the form \(Tx=gx\) in the framework of metric space. When T is a non-self mapping and g is a self-mapping, it may cause the non-existence of a solution to the preceding equation. At this juncture, one is of course interested in computing an approximate solution \(x^*\) in the space such that \(Tx^*\) is as close to \(gx^*\) as possible. To be precise, if T is from A to B and g is from A to A, where A and B are subsets of a metric space, one is concerned with the computation of a global minimizer of the mapping \(x\longrightarrow d(gx, Tx)\) which serves as a measure of closeness between Tx and gx. This paper is concerned with the resolution of the aforesaid global minimization problem if T is a proximal contraction and g is an isometry in the frameworks of fairly and proximally complete spaces.     

Dimensions of fibers of generic continuous maps

In an earlier paper Buczolich, Elekes, and the author described the Hausdorff dimension of the level sets of a generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space K by introducing the notion of topological Hausdorff dimension. Later on, the author extended the theory for maps from K to \({\mathbb {R}}^n\). The main goal of this paper is to generalize the relevant results for topological and packing dimensions and to obtain new results for sufficiently homogeneous spaces K even in the case case of Hausdorff dimension. Let K be a compact metric space and let us denote by \(C(K,{\mathbb {R}}^n)\) the set of continuous maps from K to \({\mathbb {R}}^n\) endowed with the maximum norm. Let \(\dim _{*}\) be one of the topological dimension \(\dim _T\), the Hausdorff dimension \(\dim _H\), or the packing dimension \(\dim _P\). Define

$$\begin{aligned} d_{*}^n(K)=\inf \left\{ \dim _{*}(K{\setminus } F): F\subset K \text { is } \sigma \text {-compact with } \dim _T F<n\right\} . \end{aligned}$$

We prove that \(d^n_{*}(K)\) is the right notion to describe the dimensions of the fibers of a generic continuous map \(f\in C(K,{\mathbb {R}}^n)\). In particular, we show that \(\sup \{\dim _{*}f^{-1}(y): y\in {\mathbb {R}}^n\} =d^n_{*}(K)\) provided that \(\dim _T K\ge n\), otherwise every fiber is finite. Proving the above theorem for packing dimension requires entirely new ideas. Moreover, we show that the supremum is attained on the left hand side of the above equation. Assume \(\dim _T K\ge n\). If K is sufficiently homogeneous, then we can say much more. For example, we prove that \(\dim _{*}f^{-1}(y)=d^n_{*}(K)\) for a generic \(f\in C(K,{\mathbb {R}}^n)\) for all \(y\in {{\mathrm{int}}}f(K)\) if and only if \(d^n_{*}(U)=d^n_{*}(K)\) or \(\dim _T U<n\) for all open sets \(U\subset K\). This is new even if \(n=1\) and \(\dim _{*}=\dim _H\). It is known that for a generic \(f\in C(K,{\mathbb {R}}^n)\) the interior of f(K) is not empty. We augment the above characterization by showing that \(\dim _T \partial f(K)=\dim _H \partial f(K)=n-1\) for a generic \(f\in C(K,{\mathbb {R}}^n)\). In particular, almost every point of f(K) is an interior point. In order to obtain more precise results, we use the concept of generalized Hausdorff and packing measures, too.

     

Algebras of Convolution-Type Operators with Piecewise Slowly Oscillating Data on Weighted Lebesgue Spaces

Let \({\mathcal B}_{p,w}\) be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space \(L^p(\mathbb {R},w)\), where \(p\in (1,\infty )\) and w is a Muckenhoupt weight. We study the Banach subalgebra \(\mathfrak {A}_{p,w}\) of \({\mathcal B}_{p,w}\) generated by all multiplication operators aI (\(a\in \mathrm{PSO}^\diamond \)) and all convolution operators \(W^0(b)\) (\(b\in \mathrm{PSO}_{p,w}^\diamond \)), where \(\mathrm{PSO}^\diamond \subset L^\infty (\mathbb {R})\) and \(\mathrm{PSO}_{p,w}^\diamond \subset M_{p,w}\) are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of \(\mathbb {R}\cup \{\infty \}\), and \(M_{p,w}\) is the Banach algebra of Fourier multipliers on \(L^p(\mathbb {R},w)\). For any Muckenhoupt weight w, we study the Fredholmness in the Banach algebra \({\mathcal Z}_{p,w}\subset \mathfrak {A}_{p,w}\) generated by the operators \(aW^0(b)\) with slowly oscillating data \(a\in \mathrm{SO}^\diamond \) and \(b\in \mathrm{SO}^\diamond _{p,w}\). Then, under some condition on the weight w, we complete constructing a Fredholm symbol calculus for the Banach algebra \(\mathfrak {A}_{p,w}\) in comparison with Karlovich and Loreto Hernández (Integr. Equations Oper. Theory 74:377–415, 2012) and Karlovich and Loreto Hernández (Integr. Equations Oper. Theory 75:49–86, 2013) and establish a Fredholm criterion for the operators \(A\in \mathfrak {A}_{p,w}\) in terms of their symbols. A new approach to determine local spectra is found.     

The Tracial Topological Rank of Extensions of $$C^*$$-Algebras

Let \( 0 \xrightarrow {}J \xrightarrow { } A \xrightarrow { }B \xrightarrow {} 0\) be an extension of \(C^*\)-algebras. Suppose that both J and B have tracial rank no more than one. It is shown that A has tracial topological rank no more than one whenever it is a quasidiagonal extension, and A has property \((P_1)\) if the extension is tracially quasidiagonal.     

The primitive idempotents and weight distributions of irreducible constacyclic codes

Let \({\mathbb {F}}_q\) be a finite field with q elements such that \(l^v||(q^t-1)\) and \(\gcd (l,q(q-1))=1\), where l, t are primes and v is a positive integer. In this paper, we give all primitive idempotents in a ring \(\mathbb F_q[x]/\langle x^{l^m}-a\rangle \) for \(a\in {\mathbb {F}}_q^*\). Specially for \(t=2\), we give the weight distributions of all irreducible constacyclic codes and their dual codes of length \(l^m\) over \({\mathbb {F}}_q\).     

Hilbert $$C^*$$-modules as a subcategory of operator systems and injectivity

In this paper, we study a category whose objects are Hilbert \(C^*\)-modules and whose morphisms are completely semi-\(\phi \)-maps. We give a characterization of injective objects in this category. In fact, we investigate extendability of completely semi-\(\phi \)-maps on Hilbert \(C^*\)-modules, leading to an analog of the Arveson’s extension theorem for completely semi-\(\phi \)-maps (in contrast with \(\phi \)-maps). This theorem together with previous results suggest that the completely semi-\(\phi \)-maps are proper generalizations of the completely positive maps.     

Structure theorem of the generator of a norm continuous completely positive semigroup: an alternative proof using Bures distance

Here we present an alternative proof using Bures distance that the generator L of a norm continuous completely positive semigroup acting on a \(C^*\)-algebra \({\mathcal {B}}\subset \mathcal B(H)\) has the form \( L(b) = \Psi (b) + k^*b+bk\), \(b\in {\mathcal {B}}\) for some completely positive map \(\Psi :{\mathcal {B}}\rightarrow {\mathcal {B}}(H)\) and \(k\in {\mathcal {B}}(H)\).