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Based on the success of a well-known method for solving higher order linear differential equations, a study of two of the most important mathematical features of that method, viz. the null spaces and commutativity of the product of…
I give a new derivation of the Explicit Formula for an arbitrary number field and abelian Dirichlet-Hecke character, which treats all primes in exactly the same way, whether they are discrete or archimedean, and also ramified or not. This…
A natural generalization of Krein's theorem to a pair of commuting tuples $\left(H_1^0,H_2^0\right)$ and $\left(H_1,H_2\right)$ of bounded self-adjoint operators in a separable Hilbert space $\mathcal{H}$ with $H_j-H_j^0 = V_j\in…
This paper primarily investigates spectral properties of symmetric tensor products of Hilbert-space operators. For a unilateral weighted shift operator $S_w$, we present an algorithm to compute the point spectrum of its symmetric and…
Given Hilbert spaces $H_1,H_2,H_3$, we consider bilinear maps defined on the cartesian product $S^2(H_2,H_3)\times S^2(H_1,H_2)$ of spaces of Hilbert-Schmidt operators and valued in either the space $B(H_1,H_3)$ of bounded operators, or in…
In this paper, we generalize the Riemann-Liouville differential and integral operators on the space of Henstock-Kurzweil integrable distributions, $D_{HK}$. We obtain new fundamental properties of the fractional derivative and integral, a…
We present some reverse Young-type inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with operator means. More precisely, we show that if $A, B\in {\mathfrak…
This paper provides a method to study the non-negativity of certain linear operators, from other operators with similar spectral properties. If these new operators are formally self-adjoint and non-negative, we can study the complex powers…
We give a review of results on the operator-norm convergence of the Trotter product formula on Hilbert and Banach spaces, which is focused on the problem of its convergence rates. Some recent results concerning evolution semigroups are…
We study the Birman-Schwinger operator for a self-adjoint realisation of the one-dimensional Hamiltonian with the Coulomb potential. We study both the case in which this Hamiltonian is defined on the whole real line and when it is only…
This paper explores the long journey from projective tensor products of a pair of Banach spaces, passing through the definition of nuclear operators still on the realm of projective tensor products, to the of notion of trace-class operators…
Motivated by the study of H\"ormander's sums-of-squares operators and their generalizations, we define the convolution algebra of transverse distributions associated to a singular foliation. We prove that this algebra is represented as…
This paper investigates spectral properties of certain classes of positive operators originated from different matrices appeared in linear complementarity problem. These positive operators play a crucial role in various areas of mathematics…
We study analytic models of operators of class $C_{\cdot 0}$ with natural positivity assumptions. In particular, we prove that for an $m$-hypercontraction $T \in C_{\cdot 0}$ on a Hilbert space $\mathcal{H}$, there exists a Hilbert space…
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…
We compute the Selberg trace formula for Hecke operators (also called the trace formula for modular correspondences) in the context of cocompact Kleinian groups with finite-dimentional unitary representations. We give some applications to…
We present a formula for vector-valued modular forms, expressing the value of the Hilbert-polynomial of the module of holomorphic forms evaluated at specific arguments in terms of traces of representation matrices, restricting the weight…
We consider an action of the real line on a C*-algebra for which there is a centre-valued invariant trace. We define a family of Toeplitz operators with symbols in the original algebra. When the symbol is invertible, the Toeplitz operator…
We study positive operator decompositions associated with rooted trees of orthogonal projections. In this sense, the refinement tree induces an ``MRA in $B\left(H\right)_{+}$''. To each node we assign a positive content operator, and these…
Given a commutative ring with identity $R$, many different and interesting operations can be defined over the set $H_R$ of sequences of elements in $R$. These operations can also give $H_R$ the structure of a ring. We study some of these…