We define the energy functional $E_{f,A}$ for a bounded linear operator $A$ acting on a Hilbert space $\mathcal{H}$ through a newly defined non-Euclidean metric $g_f\left(z\right)|dz{|}^2$ on its resolvent set $\rho (A)$, where the vector $f\in \mathcal{H}$. We investigate the extremal values of $E_{f,A}$ with respect to the change of $f$. We conduct an in-depth study of the case when $A$ is the classical Volterra operator $V$ on $L^2([0,1])$. Our main theorem suggests a likely connection between the energy functional and the invariant subspace problem.

We establish the short exact sequences associated with the algebras generated by dual truncated Toeplitz operators on the orthogonal complement of the model space $K_u^2$, and discuss spectral properties of dual truncated Toeplitz operators.

We are concerned in this article with an existence theorem for variational inequalities in nonreflexive Banach spaces with a general coercivity condition. The variational inequalities contain multivalued generalized pseudomonotone mappings and convex functionals, the nonreflexive Banach spaces form a complementary system, and the coercivity condition involves both the mapping and the functional. As an application, we study second-order elliptic variational inequalities with multivalued lower-order terms in general Orlicz–Sobolev spaces.

We study Toeplitz operators on Hilbert spaces of holomorphic functions on algebraic varieties, the generalized Kepler varieties defined in Jordan theoretic terms. Using the fine analysis of the reproducing kernel function, we construct and classify the irreducible representations of the $C^{\ast }$-algebra generated by these operators (with smooth symbols). The limit behavior of Bergman-type measures under peaking functions is of special importance.

In resonance with a recent geometric framework proposed by Douglas and Yang, we develop a functional model for certain linear bounded operators with rank-one self-commutator acting on a Hilbert space. By taking advantage of the refined existing theory of the principal function of a hyponormal operator, we transfer the whole action outside the spectrum, on the resolvent of the underlying operator, localized at a distinguished vector. The whole construction turns out to rely on an elementary algebra body involving analytic multipliers and Cauchy transforms. We propose a natural field theory interpretation of the resulting resolvent functional model.

We characterize the unitary equivalence of $2$-isometric operators satisfying the so-called kernel condition. This relies on a model for such operators built on operator-valued unilateral weighted shifts and on a characterization of the unitary equivalence of operator-valued unilateral weighted shifts in a fairly general context. We also provide a complete system of unitary invariants for $2$-isometric weighted shifts on rooted directed trees satisfying the kernel condition. This is formulated purely in the language of graph theory—namely, in terms of certain generation branching degrees. Finally, we study the membership of the Cauchy dual operators of $2$-isometries in classes $C_{0\cdot }$ and $C_{\cdot 0}$.

In this paper, we obtain a sufficient condition for Toeplitz operators with locally integrable symbols to be bounded and compact on the Bergman space $L_a^p$ with $1<p<\infty$ in terms of generalized Carleson squares.

Using a factorization theorem of Douglas, we prove functional characterizations of trace spaces $H^s(\partial \Omega )$ involving a family of positive self-adjoint operators. Our method is based on the use of a suitable operator by taking the trace on the boundary $\partial \Omega$ of a bounded Lipschitz domain $\Omega \subset {\mathbb{R}}^d$ and applying Moore–Penrose pseudoinverse properties together with a special inner product on $H^1(\Omega )$. We also establish generalized results of the Moore–Penrose pseudoinverse.

We introduce the notion of $\overrightarrow{s}$ -numbers for operators in Banach couples. We investigate variants of the approximation, Gelfand, and Kolmogorov numbers. In particular, we derive upper estimates of these numbers for operators between spaces generated by interpolation functors on Banach couples satisfying interpolation variants of approximation properties. We also study two-sided interpolation of entropy numbers.

We study the dependence of singularities of scattering matrices of the abstract wave equation on the choice of asymptotically equivalent outgoing/incoming subspaces. We apply the obtained results to the radial wave equation with nonlocal potential. In the latter case, the concept of associated inner function—introduced by Douglas, Shapiro, and Shields in 1970—plays an essential role.

We determine the solid hull for $2<p<\infty$ and the solid core for $1<p<2$ of weighted Bergman spaces $A_{\mu }^p,1<p<\infty$, of analytic functions on the disk and on the whole complex plane, for a very general class of nonatomic positive bounded Borel measures $\mu$. New examples are presented. Moreover, we show that the space $A_{\mu }^p,1<p<\infty$, is solid if and only if the monomials are an unconditional basis of this space.

For shift-like commuting tuples $T\in B(H{)}^n$ on graded Hilbert spaces $H$, we show that each homogeneous invariant subspace $M$ of $T$ has finite index and is generated by its wandering subspace. Under suitable conditions on the grading $(H_k{)}_{k\ge 0}$ of $H$, the algebraic direct sum $\tilde{M}={\oplus }_{k\ge 0}M\cap H_k$ becomes a finitely generated module over the polynomial ring $\mathbb{C}\left[z\right]$ in $n$ complex variables. We show that the wandering subspace $W_T\left(M\right)$ of $M$ is contained in $\tilde{M}$ and that each linear basis of $W_T\left(M\right)$ forms a minimal set of generators for $\tilde{M}$. We describe an algorithm that transforms each set of homogeneous generators of $\tilde{M}$ into a minimal set of generators and can be used to compute minimal sets of generators for homogeneous ideals $I\subset \mathbb{C}\left[z\right]$. We prove that each finitely generated $\gamma$-graded commuting row contraction $T\in B(H{)}^n$ admits a finite weak resolution in the sense of Arveson or of Douglas and Misra.