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The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that the time evolution is…

We investigate composition operators $C_{\Phi}$ on the Hardy-Smirnov space $H^{2}(\Omega)$ induced by analytic self-maps $\Phi$ of an open simply connected proper subset $\Omega$ of the complex plane. When the Riemann map…

Given a *-homomorphism $\sigma: C(M)\to \mathscr{L}(\mathcal{H})$ on a Hilbert space $\mathcal{H}$ for a compact metric space $M$, a projection $P$ onto a subspace $\mathcal{P}$ in $\mathcal{H}$ is said to be essentially normal relative to…

The Hardy space H^2(R) for the upper half plane together with a unimodular function group representation u(\lambda) = \exp(i(\lambda_1\psi_1 + ... + \lambda_n\psi_n)) for \lambda in R^n, gives rise to a manifold M of orthogonal projections…

Let $f:{\mathbb R}_+\mapsto {\mathbb R}$ be a smooth function with $f(0)=0.$ A problem of estimation of a functional $\tau_f(\Sigma):= {\rm tr}(f(\Sigma))$ of unknown covariance operator $\Sigma$ in a separable Hilbert space ${\mathbb H}$…

We study various operator homological properties of the Fourier algebra $A(G)$ of a locally compact group $G$. Establishing the converse of two results of Ruan and Xu, we show that $A(G)$ is relatively operator 1-projective if and only if…

We characterise the Kato property of a sectorial form $\mathfrak{a}$, defined on a Hilbert space $V$, with respect to a larger Hilbert space $H$ in terms of two bounded, selfadjoint operators $T$ and $Q$ determined by the imaginary part of…

Solitons in two-dimensional quantum field theory exhibit patterns of degeneracies and associated selection rules on scattering amplitudes. We develop a representation theory that captures these intriguing features of solitons. This…

We characterize Beurling quotient subspaces for pure doubly commuting isometric representations of product systems. As a consequence, we derive a concrete regular dilation theorem for a pure completely contractive covariant representation…

Let M be a connected d-dimensional complex projective manifold, and let A be a holomorphic positive Hermitian line bundle on M, with normalized curvature. Let G be a compact and connected Lie group of dimension d(G), and let T be a compact…

Let $U$ be a unitary operator defined on some infinite-dimensional complex Hilbert space ${\cal H}$. Under some suitable regularity assumptions, it is known that a local positive commutation relation between $U$ and an auxiliary…

We give a complete description of the bounded (i.e. norm continuous) unitary representations of the Fr\'echet-Lie algebra of all smooth sections, as well as of the LF-Lie algebra of compactly supported smooth sections, of a smooth Lie…

Let $G$ be a compact, connected Lie group and $T \subset G$ a maximal torus. Let $(M,\omega)$ be a monotone closed symplectic manifold equipped with a Hamiltonian action of $G$. We construct a module action of the affine nil-Hecke algebra…

We present a unified operator-theoretic framework for stochastic calculus based on the factorization (Id - E)F = {\delta}_X {\Pi}_X D_X F, valid for F_T^X-measurable F in L^2({\Omega}) when the driving process X has the representation…

Let $T$ be a pseudo-differential operator whose symbol belongs to the H\"ormander class $S^m_{\rho,\delta}$ with $0\leq \delta<1, 0< \rho\leq 1, \delta \leq \rho$ and $-(n+1)< m \leq - (n+1)(1-\rho)$. In present paper, we prove that if $b$…

A well known result of C. Cowen states that, for a symbol $\varphi \in L^{\infty }, \; \varphi \equiv \bar{f}+g \;\;(f,g\in H^{2})$, the Toeplitz operator $T_{\varphi }$ acting on the Hardy space of the unit circle is hyponormal if and only…

We establish a close link between the amenability of a unitary representation $\pi$ of a group $G$ (in the sense of Bekka) and the concentration property (in the sense of V. Milman) of the corresponding dynamical system $(\s_\pi,G)$, where…

We classify all subsets $S$ of the projective Hilbert space with the following property: for every point $\pm s_0\in S$, the spherical projection of $S\backslash\{\pm s_0\}$ to the hyperplane orthogonal to $\pm s_0$ is isometric to…

The goal of this paper is to study the structure of noncommutative weighted shifts, their properties, and to understand their role as models (up to similarity) for $n$-tuples of operators on Hilbert spaces as well as their implications to…

Let $\mathcal{H}$ be a complex, separable Hilbert space (of finite or infinite dimension), and let $\mathcal{U}(\mathcal{H})$ denote the group of unitary operators on $\mathcal{H}$. A symmetry is, by definition, a unitary operator $J$ with…