If $\mu $ is a smooth measure supported on a real-analytic submanifold of ${\mathbb {R}}^{2n}$ which is not contained in any affine hyperplane, then the Weyl transform of $\mu $ is a compact operator.

Any Lipschitz map $f : M \to N$ between two pointed metric spaces may be extended in a unique way to a bounded linear operator $\widehat {f} : \mathcal {F}(M) \to \mathcal {F}(N)$ between their corresponding Lipschitz-free spaces. In this paper, we give a necessary and sufficient condition for $\widehat {f}$ to be compact in terms of metric conditions on $f$. This extends a result by A. Jiménez-Vargas and M. Villegas-Vallecillos in the case of non-separable and unbounded metric spaces. After studying the behaviour of weakly convergent sequences made of finitely supported elements in Lipschitz-free spaces, we also deduce that $\widehat {f}$ is compact if and only if it is weakly compact.

In this paper, we discuss the properties of the embedding operator $i_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6EC}}:M_{\unicode[STIX]{x1D6EC}}^{\infty }{\hookrightarrow}L^{\infty }(\unicode[STIX]{x1D707})$, where $\unicode[STIX]{x1D707}$ is a positive Borel measure on $[0,1]$ and $M_{\unicode[STIX]{x1D6EC}}^{\infty }$ is a Müntz space. In particular, we compute the essential norm of this embedding. As a consequence, we recover some results of the first author. We also study the compactness (resp. weak compactness) and compute the essential norm (resp. generalized essential norm) of the embedding $i_{\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2}}:L^{\infty }(\unicode[STIX]{x1D707}_{1}){\hookrightarrow}L^{\infty }(\unicode[STIX]{x1D707}_{2})$, where $\unicode[STIX]{x1D707}_{1}$, $\unicode[STIX]{x1D707}_{2}$ are two positive Borel measures on [0, 1] with $\unicode[STIX]{x1D707}_{2}$ absolutely continuous with respect to $\unicode[STIX]{x1D707}_{1}$.

We study the Bishop-Phelps-Bollobàs property $\left( \text{BPBp} \right)$ for compact operators. We present some abstract techniques that allow us to carry the $\text{BPBp}$ for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let $X$ and $Y$ be Banach spaces. If $\left( {{c}_{0}},Y \right)$ has the $\text{BPBp}$ for compact operators, then so do $\left( {{C}_{0}}\left( L \right),Y \right)$ for every locally compact Hausdorff topological space $L$ and $\left( X,\,Y \right)$ whenever ${{X}^{*}}$ is isometrically isomorphic to ${{\ell }_{1}}$. If ${{X}^{*}}$ has the Radon-Nikodým property and $\left( {{\ell }_{1}}\left( X \right),\,Y \right)$ has the $\text{BPBp}$ for compact operators, then so does $\left( {{L}_{1}}\left( \mu ,X \right),\,\,Y \right)$ for every positive measure $\mu $; as a consequence, $\left( {{L}_{1}}\left( \mu ,X \right),\,\,Y \right)$ has the $\text{BPBp}$ for compact operators when $X$ and $Y$ are finite-dimensional or $Y$ is a Hilbert space and $X={{c}_{0}}$ or $X={{L}_{p}}\left( v \right)$ for any positive measure $v$ and $1\,<\,p\,<\,\infty $. For $1\,\le p\,<\,\infty$, if $\left( X,{{l}_{p}}(Y) \right)$ has the $\text{BPBp}$ for compact operators, then so does $\left( X,{{L}_{p}}\left( \mu ,\,Y \right) \right)$ for every positive measure $\mu $ such that ${{L}_{1}}\left( \mu \right)$ is infinite-dimensional. If $\left( X,\,Y \right)$ has the $\text{BPBp}$ for compact operators, then so do $\left( X,\,{{L}_{\infty }}\left( \mu ,\,\,Y \right) \right)$ for every $\sigma $-finite positive measure $\mu $ and $\left( X,\,C\left( K,\,Y \right) \right)$ for every compact Hausdorff topological space $K$.

The Birkhoff orthogonality has been recently intensively studied in connection with the geometry of Banach spaces and operator theory. The main aim of this paper is to characterize the Birkhoff orthogonality in ${\mathcal{L}}(X;Y)$ under the assumption that ${\mathcal{K}}(X;Y)$ is an $M$-ideal in ${\mathcal{L}}(X;Y)$. Moreover, we survey the known results, as well as giving some new and more general ones. Furthermore, we characterize an approximate Birkhoff orthogonality in ${\mathcal{K}}(X;Y)$.

We analyze domination properties and factorization of operators in Banach spaces through subspaces of $L^{1}$-spaces. Using vector measure integration and extending classical arguments based on scalar integral bounds, we provide characterizations of operators factoring through subspaces of $L^{1}$-spaces of finite measures. Some special cases involving positivity and compactness of the operators are considered.

Let ${{T}_{\Omega }}$ be the singular integral operator with kernel $\left( \Omega \left( x \right) \right)/{{\left| x \right|}^{n}}$, where $\Omega $ is homogeneous of degree zero, has mean value zero, and belongs to ${{L}^{q}}\left( {{S}^{n-1}} \right)$ for some $q\,\in \,\left( 1,\,\infty \right]$. In this paper, the authors establish the compactness on weighted ${{L}^{p}}$ spaces and the Morrey spaces, for the commutator generated by $\text{CMO}\left( {{\mathbb{R}}^{n}} \right)$ function and ${{T}_{\Omega }}$. The associated maximal operator and the discrete maximal operator are also considered.

An investigation is made of the continuity, the compactness and the spectrum of the Cesàro operator $\mathsf{C}$ when acting on the weighted Banach sequence spaces $\ell _{p}(w)$, $1<p<\infty$, for a positive decreasing weight $w$, thereby extending known results for $\mathsf{C}$ when acting on the classical spaces $\ell _{p}$. New features arise in the weighted setting (for example, existence of eigenvalues, compactness) which are not present in $\ell _{p}$.

Let $b\,\in \,\text{BMO}\left( {{\mathbb{R}}^{n}} \right)$ and ${{T}_{\Omega }}$ be the singular integral operator with kernel $\Omega \left( x \right)/{{\left| x \right|}^{n}}$, where $\Omega$ is homogeneous of degree zero, integrable, and has mean value zero on the unit sphere ${{S}^{n-1}}$. In this paper, using Fourier transform estimates and approximation to the operator ${{T}_{\Omega }}$ by integral operators with smooth kernels, it is proved that if $b\,\in \,\text{CMO}\left( {{\mathbb{R}}^{n}} \right)$ and $\Omega$ satisfies certain minimal size condition, then the commutator generated by $b$ and ${{T}_{\Omega }}$ is a compact operator on ${{L}^{p}}\left( {{\mathbb{R}}^{n}} \right)$ for appropriate index $p$. The associated maximal operator is also considered.

In this paper we prove a structure theorem for the class of $\mathcal{AN}$-operators between separable, complex Hilbert spaces which is similar to that of the singular value decomposition of a compact operator. Apart from this, we show that a bounded operator is $\mathcal{AN}$ if and only if it is either compact or a sum of a compact operator and scalar multiple of an isometry satisfying some condition. We obtain characterizations of these operators as a consequence of this structure theorem and deduce several properties which are similar to those of compact operators.

We study the boundedness and compactness of Toeplitz operators Ta on Bergman spaces , 1 < p < ∞. The novelty is that we allow distributional symbols. It turns out that the belonging of the symbol to a weighted Sobolev space of negative order is sufficient for the boundedness of Ta. We show the natural relation of the hyperbolic geometry of the disc and the order of the distribution. A corresponding sufficient condition for the compactness is also derived.

We deal with the dual Banach algebras for a locally compact group G. We investigate compact left multipliers on , and prove that the existence of a compact left multiplier on is equivalent to compactness of G. We also describe some classes of left completely continuous elements in .

We give an example of a Banach space X such that K (X, X) is not an ideal in K (X, X**). We prove that if z* is a weak* denting point in the unit ball of Z* and if X is a closed subspace of a Banach space Y, then the set of norm-preserving extensions H B(x* ⊗ z*) ⊆ (Z*, Y)* of a functional x* ⊗ Z* ∈ (Z ⊗ X)* is equal to the set H B(x*) ⊗ {z*}. Using this result, we show that if X is an M-ideal in Y and Z is a reflexive Banach space, then K (Z, X) is an M-ideal in K(Z, Y) whenever K (Z, X) is an ideal in K (Z, Y). We also show that K (Z, X) is an ideal (respectively, an M-ideal) in K (Z, Y) for all Banach spaces Z whenever X is an ideal (respectively, an M-ideal) in Y and X * has the compact approximation property with conjugate operators.

In this paper we consider collections of compact operators on a real or complex Banach space including linear operators on finite-dimensional vector spaces. We show that such a collection is simultaneously triangularizable if and only if it is arbitrarily close to a simultaneously triangularizable collection of compact operators. As an application of these results we obtain an invariant subspace theorem for certain bounded operators. We further prove that in finite dimensions near reducibility implies reducibility whenever the ground field is $\mathbb{R}$ or $\mathbb{C}$.

The ${{Q}_{p}}$ spaces coincide with the Bloch space for $p\,>\,1$ and are subspaces of $\text{BMOA}$ for $0\,<\,p\,\le \,1$. We obtain lower and upper estimates for the essential norm of a composition operator from the Bloch space into ${{Q}_{p}}$, in particular from the Bloch space into $\text{BMOA}$.

Every weakly compact composition operator between weighted Banach spaces $H_{v}^{\infty }$ of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space $H_{v}^{\infty }$ are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces.