Multipliers on homogeneous Banach spaces with respect to Jacobi polynomials

A bounded linear operator is called multiplier with respect to Jacobi polynomials if and only if it commutes with all Jacobi translation operators on $[-1,1]$ . Multipliers on homogeneous Banach spaces on $[-1,1]$ determined by the Jacobi translation operator are introduced and studied. First we prove four equivalent characterizations of a multiplier for an arbitrary homogeneous Banach spaces $B$ on $[-1,1]$ . One of them implies the existence of an algebra isomorphism from the set of all multipliers on $B$ into the set of all pseudomeasures. Further, we study multipliers on specific examples of homogeneous Banach spaces on $[-1,1]$ . Amongst others, multipliers on the Wiener algebra, on the Beurling space and on Sobolev spaces are analyzed. We obtain that the multiplier spaces of the Wiener algebra, the Beurling space and of all Sobolev spaces are isometric isomorphic to each other. Furthermore, these multiplier spaces are all isometric isomorphic to the set of all pseudomeasures.     

Wiener’s lemma for singular integral operators of Bessel potential type

In this paper, we introduce an algebra of singular integral operators containing Bessel potentials of positive order, and show that the corresponding unital Banach algebra is an inverse-closed Banach subalgebra of ${\mathcal {B}} (L^q_w)$ , the Banach algebra of all bounded operators on the weighted space $L_w^q$ , for all $1\le q<\infty $ and Muckenhoupt $A_q$ -weights $w$ .     

A commutant realization of Odake’s algebra

The bcβγ-system $ \mathcal{W} $ of rank 3 has an action of the affine vertex algebra $ {V_0}\left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ , and the commutant vertex algebra $ \mathcal{C}=\mathrm{Com}\left( {{V_0}\left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right),\mathcal{W}} \right) $ contains copies of V _?3/2 $ \left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ and Odake’s algebra $ \mathcal{O} $ . Odake’s algebra is an extension of the N = 2 super-conformal algebra with c = 9, and is generated by eight fields which close nonlinearly under operator product expansions. Our main result is that V _?3/2 $ \left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ and $ \mathcal{O} $ form a Howe pair (i.e., a pair of mutual commutants) inside $ \mathcal{C} $ . More generally, any finite-dimensional representation of a Lie algebra $ \mathfrak{g} $ gives rise to a similar Howe pair, and this example corresponds to the adjoint representation of $ \mathfrak{s}{{\mathfrak{l}}_2} $ .     

Norm Transitive Semigroups on Operator Algebra

Let $\mathcal A $ be a semigroup of bounded linear operators on the Banach algebra $B(X)$ for a separable Banach space $X$ . We show the transitivity of $\mathcal A $ with the operator norm topology, implies hypercyclicity with the strong operator topology (SOT) while the converse may not be true. As a consequence, SOT-transitive semigroup of left multiplication operators on $B(X)$ is characterized.     

Boundedness and Compactness of Pseudodifferential Operators with Non-Regular Symbols on Weighted Lebesgue Spaces

Applying the boundedness on weighted Lebesgue spaces of the maximal singular integral operator S _* related to the Carleson?CHunt theorem on almost everywhere convergence, we study the boundedness and compactness of pseudodifferential operators a(x, D) with non-regular symbols in ${L^\infty(\mathbb{R}, V(\mathbb{R})), PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ and ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ on the weighted Lebesgue spaces ${L^p(\mathbb{R},w)}$ , with 1?< p <? ?? and ${w\in A_p(\mathbb{R})}$ . The Banach algebras ${L^\infty(\mathbb{R}, V(\mathbb{R}))}$ and ${PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ consist, respectively, of all bounded measurable or piecewise continuous ${V(\mathbb{R})}$ -valued functions on ${\mathbb{R}}$ where ${V(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded total variation, and the Banach algebra ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ consists of all Lipschitz ${V_d(\mathbb{R})}$ -valued functions of exponent ${\gamma \in (0,1]}$ on ${\mathbb{R}}$ where ${V_d(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded variation on dyadic shells. Finally, for the Banach algebra ${\mathfrak{A}_{p,w}}$ generated by all pseudodifferential operators a(x, D) with symbols ${a(x, \lambda) \in PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ on the space ${L^p(\mathbb{R}, w)}$ , we construct a non-commutative Fredholm symbol calculus and give a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ .     

Algebras of Convolution Type Operators with Piecewise Slowly Oscillating Data. I: Local and Structural Study

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 ( ${a \in PSO^{\diamond}}$ ) and all convolution operators W ⁰(b) ( ${b \in PSO_{p,w}^{\diamond}}$ ), where ${PSO^{\diamond} \subset L^{\infty}(\mathbb{R})}$ and ${PSO_{p,w}^{\diamond} \subset M_{p,w}}$ 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 ${L^{p}(\mathbb{R}, w)}$ . Under some conditions on the Muckenhoupt weight w, we construct a Fredholm symbol calculus for the Banach algebra ${\mathfrak{A}_{p,w}}$ and establish a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ in terms of their Fredholm symbols. To study the Banach algebra ${\mathfrak{A}_{p,w}}$ we apply the theory of Mellin pseudodifferential operators, the Allan–Douglas local principle, the two idempotents theorem and the method of limit operators. The paper is divided in two parts. The first part deals with the local study of ${\mathfrak{A}_{p,w}}$ and necessary tools for studying local algebras.     

Duhamel convolution product in the setting of quantum calculus

In this paper, we introduce the notions of the q-Duhamel product and q-integration operator. We prove that the classical Wiener algebra \(W_+(\mathbb {D})\) of all analytic functions on the unit disc \(\mathbb {D}\) of the complex plane \(\mathbb {C}\) with absolutely convergent Taylor series extended to the boundary is a Banach algebra with respect to the q-Duhamel product. We also describe the cyclic vectors of the q-integration operator on \(W_+(\mathbb {D})\) and characterize its commutant in terms of the q-Duhamel product operators.     

Factorizing Kernel Operators

Consider an operator ${T: X(\mu) \rightarrow Y(\mu)}$ between Banach function spaces having adequate order continuity and Fatou properties. Assume that T can be factorized through a Banach space as ${T = S \circ R}$ , where R and the adjoint of S are p-th power and q-th power factorable, respectively. Then a canonical factorization scheme can be given for T. We show that it provides a tool for analyzing T that becomes specially useful for the case of kernel operators. In particular, we show that this square factorization scheme for T is equivalent to some inequalities for the bilinear form defined by T. Kernel operators are studied from this point of view.     

Analytic Toeplitz algebras and the Hilbert transform associated with a subdiagonal algebra

Let M be aσ-finite von Neumann algebra and let AM be a maximal subdiagonal algebra with respect to a faithful normal conditional expectationΦ.Based on the Haagerup’s noncommutative Lpspace Lp(M)associated with M,we consider Toeplitz operators and the Hilbert transform associated with A.We prove that the commutant of left analytic Toeplitz algebra on noncommutative Hardy space H2(M)is just the right analytic Toeplitz algebra.Furthermore,the Hilbert transform on noncommutative Lp(M)is shown to be bounded for 1p∞.As an application,we consider a noncommutative analog of the space BMO and identify the dual space of noncommutative H1(M)as a concrete space of operators.     

Algebras of Convolution Type Operators with Piecewise Slowly Oscillating Data. II: Local Spectra and Fredholmness

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 ${\mathfrak{U}_{p,w}}$ of ${\mathcal{B}_{p,w}}$ generated by all multiplication operators aI ( ${a\in PSO^\diamond}$ ) and all convolution operators W ⁰(b) ( ${b\in PSO_{p,w}^\diamond}$ ), where ${PSO^\diamond\subset L^\infty(\mathbb{R})}$ and ${PSO_{p,w}^\diamond\subset M_{p,w}}$ 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 ${L^p(\mathbb{R},w)}$ . Under some conditions on the Muckenhoupt weight w, using results of the local study of ${\mathfrak{U}_{p,w}}$ obtained in the first part of the paper and applying the theory of Mellin pseudodifferential operators and the two idempotents theorem, we now construct a Fredholm symbol calculus for the Banach algebra ${\mathfrak{U}_{p,w}}$ and establish a Fredholm criterion for the operators ${A\in\mathfrak{U}_{p,w}}$ in terms of their Fredholm symbols. In four partial cases we obtain for ${\mathfrak{U}_{p,w}}$ more effective results.     

Spectral Properties for Matrix Algebras

We consider Banach algebras of infinite matrices defined in terms of a weight measuring the off-diagonal decay of the matrix entries. If a given matrix $A$ is invertible as an operator on $\ell ^2$ we analyze the decay of its inverse matrix entries in the case where the matrix algebra is not inverse closed in ${\mathcal B} (\ell ^2),$ the Banach algebra of bounded operators on $\ell ^2.$ To this end we consider a condition on sequences of weights which extends the notion of GRS-condition. Finally we focus on the behavior of inverses of pseudodifferential operators whose Weyl symbols belong to weighted modulation spaces and the weights lack the GRS condition.     

Quasi-Banach operator ideals with a very strange trace

As shown by S. Lord, F. Sukochev, and D. Zanin (see [7]), the theory of singular traces is well understood for operators on the Hilbert space. The situation turns out to be completely different in the Banach space setting. Indeed, quite strange phenomena may occur. We will construct quasi-Banach operator ideals ${\mathfrak A}$ A with seemingly contradictory properties: On the one hand, ${\mathfrak A}$ A supports a continuous trace τ that vanishes at all finite rank operators, which means that τ is singular. On the other hand, ${\mathfrak A}$ A contains the identity map I _Z of an infinite-dimensional Banach space Z and τ (I _Z ) =  1. This implies that there exist operators ${T \in \mathfrak A (Z)}$ T ∈ A ( Z ) such that ${\tau (T^n) = 1}$ τ ( T n ) = 1 for ${n = 1,2,{\dots} \;}$ n = 1 , 2 , ? , which is impossible for singular traces in the case of a Hilbert space. As most counterexamples, the new operator ideals have no useful application. They provide, however, a deeper insight into the philosophy of traces.     

Generalized Hille-Phillips type functional calculus for multiparameter semigroups

For generators of n-parameter strongly continuous operator semigroups in a Banach space, we construct a Hille-Phillips type functional calculus, the symbol class of which consists of analytic functions from the image of the Laplace transform of the convolution algebra of temperate distributions supported by the positive cone ? ₊ ⁿ . The image of such a calculus is described with the help of the commutant of the semigroup of shifts along the cone. The differential properties of the calculus and some examples are presented.     

Isometric factorization of weakly compact operators and the approximation property

Using an isometric version of the Davis, Figiel, Johnson, and Pe?czyński factorization of weakly compact operators, we prove that a Banach spaceX has the approximation property if and only if, for every Banach spaceY, the finite rank operators of norm ≤1 are dense in the unit ball ofW(Y,X), the space of weakly compact operators fromY toX, in the strong operator topology. We also show that, for every finite dimensional subspaceF ofW(Y,X), there are a reflexive spaceZ, a norm one operatorJ:Y→Z, and an isometry Φ :F →W(Y,X) which preserves finite rank and compact operators so thatT=Φ(T) oJ for allT∈F. This enables us to prove thatX has the approximation property if and only if the finite rank operators form an ideal inW(Y,X) for all Banach spacesY.     

Additive maps preserving operators of inner local spectral radius zero

Let \({\mathcal {B}}(X)\) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space \(X\) . We determine the form of surjective additive maps \(\varphi :{\mathcal {B}}(X)\rightarrow {{\mathcal {B}}(X)}\) which preserve operators of inner local spectral radius zero at points of \(X\) .     

Pseudodifferential Operators Associated with a Semigroup of Operators

Related to a semigroup of operators on a metric measure space, we define and study pseudodifferential operators (including the setting of Riemannian manifolds, fractals, graphs etc.). Boundedness on L ^p for pseudodifferential operators of order 0 is proved. We mainly focus on symbols belonging to the class $S^{0}_{1,\delta}$ for δ∈[0,1). For the limit class $S^{0}_{1,1}$ , we describe some results by restricting our attention to the case of a sub-Laplacian operator on a Riemannian manifold.     

Symmetry and quasi-centrality of the operator space projective tensor product

For C*-algebras A and B, the operator space projective tensor product ${A\widehat{\otimes}B}$ and the Banach space projective tensor product ${A\otimes_{\gamma}B}$ are shown to be symmetric. We also show that ${A\widehat{\otimes}B}$ is a weakly Wiener algebra. Finally, quasi-centrality and the unitary group of ${A\widehat{\otimes}B}$ are discussed.     

Local Definitizability of {{T^{[\ast]}T}} and {{TT^{{[\ast]}}}}

The spectral properties of two products AB and BA of possibly unbounded operators A and B in a Banach space are considered. The results are applied in the comparison of local spectral properties of the operators ${T^{[\ast]} T}$ and ${TT^{[\ast]}}$ in a Krein space. It is shown that under the assumption that both operators ${T^{[\ast]} T}$ and ${TT^{[\ast]}}$ have non-empty resolvent sets, the operator ${T^{[\ast]} T}$ is locally definitizable if and only if ${TT^{[\ast]}}$ is. In this context the critical points of both operators are compared.     

Self-adjoint commuting ordinary differential operators

In this paper we study self-adjoint commuting ordinary differential operators of rank two. We find sufficient conditions when an operator of fourth order commuting with an operator of order 4g+2 is self-adjoint. We introduce an equation on potentials V(x),W(x) of the self-adjoint operator \(L=(\partial_{x}^{2}+V)^{2}+W\) and some additional data. With the help of this equation we find the first example of commuting differential operators of rank two corresponding to a spectral curve of higher genus. These operators have polynomial coefficients and define commutative subalgebras of the first Weyl algebra.     

Supercyclicity in Spaces of Operators

A sufficient criterion for the map \({C_{A, B}(S) = ASB}\) to be supercyclic on certain algebras of operators on Banach spaces is given. If T is an operator satisfying the Supercyclicity Criterion on a Hilbert space H, then the linear map \({C_{T}(V) = TVT^*}\) is shown to be norm-supercyclic on the algebra \({\mathcal{K}(H)}\) of all compact operators, COT-supercyclic on the real subspace \({\mathcal{S}(H)}\) of all self-adjoint operators and weak*-supercyclic on \({\mathcal{L}(H)}\) of all bounded operators on H. Examples including operators of the form \({C_{B_w, F_\mu}}\) are provided, where B_w and \({F_\mu}\) are respectively backward and forward shifts on Banach sequence spaces.