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We consider the products of composition and differentiation operators on the Hardy space. We provide a complete characterization of the boundedness and compactness of these operators. Furthermore, we obtain the explicit condition for these…

We completely characterize the boundedness of the area operators from the Bergman spaces $A^p_\alpha(\mathbb{B}_ n)$ to the Lebesgue spaces $L^q(\mathbb{S}_ n)$ for all $0<p,q<\infty$. For the case $n=1$, some partial results were…

This paper is concerned with paired operators in the context of the Lebesgue Hilbert space on the unit circle and its subspace, the Hardy space. By considering when such operators commute, generalizations of the Brown--Halmos results for…

We carry on the study of Fourier integral operators of H{\"o}rmander's type acting on the spaces $(\mathcal{F}L^p)_{comp}$, $1\leq p\leq\infty$, of compactly supported distributions whose Fourier transform is in $L^p$. We show that the…

Hankel operators with anti-holomorphic symbols are studied for a large class of weighted Fock spaces on $\cn$. The weights defining these Hilbert spaces are radial and subject to a mild smoothness condition. In addition, it is assumed that…

The basic results for nonlinear operators are given. These results include nonlinear versions of classical uniform boundedness theorem and Hahn-Banach theorem. Furthermore, the mappings from a metrizable space into another normed space can…

Suppose that $\mathscr{E}$ and $\mathscr{F}$ are Hilbert $C^*$-modules. We present a power-norm $\left(\left\|\cdot\right\|^{\mathscr{E}}_n:n\in\mathbb{N}\right)$ based on $\mathscr{E}$ and obtain some of its fundamental properties. We…

Given a bounded linear operator $T$ on separable Hilbert space, we develop an approach allowing one to construct a matrix representation for $T$ having certain specified algebraic or asymptotic structure. We obtain matrix representations…

Using the approach proposed in [5] , in an infinite-dimensional separable complex Hilbert space we give abstract constructions of families $\{{\mathcal T}_z\}_{{\rm Im\,} z>0}$ of closed densely defined symmetric operators with the…

We give an algorithm to write down all conformally invariant differential operators acting between scalar functions on Minkowski space. All operators of order k are nonlinear and are functions on a finite family of functionally independent…

Let $T$ be a bounded linear operator on a Hilbert space $H$ such that \[ \alpha[T^*,T]:=\sum_{n=0}^\infty \alpha_n T^{*n}T^n\ge 0. \] where $\alpha(t)=\sum_{n=0}^\infty \alpha_n t^n$ is a suitable analytic function in the unit disc…

This paper aims to study reducible and irreducible approximation in the set $\textsl{CSO}$ of all complex symmetric operators on a separable, complex Hilbert space $\mathcal H$. When ${\rm dim} \mathcal H=\infty$, it is proved that both…

We investigate the global boundedness of Fourier integral operators with amplitudes in the general H\"ormander classes $S^{m}_{\rho, \delta}(\mathbb{R}^n)$, $\rho, \delta\in [0,1]$ and non-degenerate phase functions of arbitrary rank…

For any positive integers $n$ and $m$, $\mathbb{H}_{n,m}:=\mathbb{H}_n\times\mathbb{C}^{(m,n)}$ is called the Siegel-Jacobi space, with the Jacobi group acting on it. The Jacobi forms are defined on this space. In this article we compute…

A closed densely defined operator $ T $ on a Hilbert space $ \mathcal{H} $ is callled $M$-hyponormal if $\mathcal{D}(T) \subset \mathcal{D}(T^{*}) $ and there exists $ M > 0 $ for which $ \parallel(T-zI)^{*}x \parallel \leq M…

We suppose that $G$ is a locally compact abelian group, $Y$ is a measure space, and $H$ is a reproducing kernel Hilbert space on $G\times Y$ such that $H$ is naturally embedded into $L^2(G\times Y)$ and it is invariant under the…

Let $A$ be a positive definite operator on a Hilbert space $H$, and $|||.|||$ be a unitarily invariant norm on $B(H)$. We show that if $f$ is an operator monotone function on $(0,\infty)$ and $n\in \mathbb{N}$, then $|||D^n…

The theory of monotone operators plays a major role in modern optimization and many areas of nonlinera analysis. The central classes of monotone operators are matrices with a positive semidefinite symmetric part and subsifferential…

Using the bicomplex numbers $\mathbb{T}$ which is a commutative ring with zero divisors defined by $\mathbb{T}=\{w_0 + w_1 i_1 + w_2 i_2 + w_3 j | w_0, w_1, w_2, w_3 \in \mathbb{R}\}$ where $i_{1}^{2} = -1, i_{2}^{2} = -1, j^2 = 1, i_1 i_2…

In this work, firstly in the direct sum of Hilbert spaces of vector-functions $L^{2} (H,(-\infty,a_{1})) \oplus L^{2} (H,(a_{2},b_{2}))\oplus^{2} (H,(a_{3},+\infty))$, $- \infty<a_{1}<a_{2}<b_{2}<a_{3}<+\infty$ all normal extensions of the…