In the present paper, we prove a resolvent equation for the $\mathcal{S}$-resolvent operator in the quaternionic framework. Exploiting this resolvent equation, we find a series expansion for the $\mathcal{S}$-resolvent operator in an open neighborhood of any given quaternion belonging to the $\mathcal{S}$-resolvent set. Some consequences of the series expansion are deduced. In particular, we describe a property of the geometry of the $\mathcal{S}$-resolvent set in terms of the Cassini pseudo-metric on quaternions. The concept of vector-valued real analytic function of several variables plays a crucial role in the proof of the mentioned series expansion for the $\mathcal{S}$-resolvent operator.
The Fueter-Sce-Qian mapping theorem is a two steps procedure to extend holomorphic functions of one complex variable to quaternionic or Clifford algebra-valued functions in the kernel of a suitable generalized Cauchy-Riemann operator. Using the Cauchy formula of slice monogenic functions it is possible to give the Fueter-Sce-Qian extension theorem an integral form and to define the $\mathcal{F}$-functional calculus for $n$-tuples of commuting operators. This functional calculus is defined on the $S$-spectrum but it generates a monogenic functional calculus in the spirit of McIntosh and collabo…
Let $\mathcal{H}$ be a right quaternionic Hilbert space and let $T$ be a quaternionic normal operator with the domain $\mathcal{D}(T) \subset \mathcal{H}$. Then for a fixed unit imaginary quaternion $m$, there exists a Hilbert basis $\mathcal{N}_{m}$ of $\mathcal{H}$, a measure space $(Ω, μ)$, a unitary operator $U \colon \mathcal{H} \to L^{2}(Ω; \mathbb{H}; μ)$ and a $μ$ - measurable function $ϕ\colon Ω\to \mathbb{C}_m$ (here $\mathbb{C}_{m} = \{α+ m β; \;α, β\in \mathbb{R}\}$) such that \[ Tx = U^{*}M_ϕUx, \; \mbox{for all}\; x\in \mathcal{D}(T), \] where $M_ϕ$ is the multiplication operator…
The quaternionic spectral theorem has already been considered in the literature, see e.g. [22], [31], [32], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not specified which notion of spectrum underlies the theorem. In this paper we prove the quaternionic spectral theorem for unitary operators using the $S$-spectrum. In the case of quaternionic matrices, the $S$-spectrum coincides with the right-spectrum and so our result recovers the well known theorem for matrices. The notion of $S$-spectrum is relative…
In this article, we prove the following spectral theorem for right linear normal operators (need not to be bounded) in quaternionic Hilbert spaces: Let $T$ be an unbounded right quaternionic linear normal operator in a quaternionic Hilbert space $H$ with domain $\mathcal{D}(T)$, a right linear subspace of $H$ and fix a unit imaginary quaternion, say $m$. Then there exists a Hilbert basis $\mathcal{N}$ of $H$ and a unique quaternionic spectral measure $F$ on the $σ$- algebra of $\mathbb C_m^{+}$ (upper half plane of the slice complex plane $\mathbb C_m$) associated to $T$ such that \begin{equat…
In this paper we develop a functional calculus for bounded operators defined on quaternionic Banach spaces. This calculus is based on the notion of slice-regularity, see \cite{gs}, and the key tools are a new resolvent operator and a new eigenvalue problem.
An isometric operator $V$ in a Pontryagin space $\mathcal{H}$ is called standard, if its domain and the range are nondegenerate subspaces in $\mathcal{H}$. Generalized resolvents of standard isometric operators were described in the paper of A. Dijksma, H. Langer, and H. de Snoo in 1990. In the present paper generalized resolvents of non-standard Pontryagin space isometric operators are described. The method of the proof is based on the notion of boundary triplet of isometric operators in Pontryagin spaces. In the Hilbert space setting the notion of boundary triplet for isometric operators was…
The analogue of the Riesz-Dunford functional calculus has been introduced and studied recently as well as the theory of semigroups and groups of linear quaternionic operators. In this paper we suppose that $T$ is the infinitesimal generator of a strongly continuous group of operators $(\mathcal{Z}_T(t))_{t \in \mathbb{R}}$ and we show how we can define bounded operators $f(T)$, where $f$ belongs to a class of functions which is larger than the class of slice regular functions, using the quaternionic Laplace-Stieltjes transform. This class will include functions that are slice regular on the $S…
The spectral theory on the S-spectrum was introduced to give an appropriate mathematical setting to quaternionic quantum mechanics, but it was soon realized that there were different applications of this theory, for example, to fractional heat diffusion and to the spectral theory for the Dirac operator on manifolds. In this seminal paper we introduce the harmonic functional calculus based on the S-spectrum and on an integral representation of axially harmonic functions. This calculus can be seen as a bridge between harmonic analysis and the spectral theory. The resolvent operator of the harmon…
We consider a class of manifolds $\mathcal{M}$ obtained by taking the connected sum of a finite number of $N$-dimensional Riemannian manifolds of the form $(\mathbb{R}^{n_i}, δ) \times (\mathcal{M}_i, g)$, where $\mathcal{M}_i$ is a compact manifold, with the product metric. The case of greatest interest is when the Euclidean dimensions $n_i$ are not all equal. This means that the ends have different `asymptotic dimension', and implies that the Riemannian manifold $\mathcal{M}$ is not a doubling space. In the first paper in this series, by the first and third authors, we considered the case wh…
We consider an arbitrary metric graph, to which we glue another graph with edges of lengths proportional to $\varepsilon$, where $\varepsilon$ is a small positive parameter. On such graph, we consider a general self-adjoint second order differential operator $\mathcal{H}_\varepsilon$ with varying coefficients subject to general vertex conditions; all coefficients in differential expression and vertex conditions are supposed to be analytic in $\varepsilon$. We introduce a special operator on a certain graph obtained by rescaling the aforementioned small edges and assume that it has no embedded …
In this paper, we consider an unbounded selfadjoint operator $A$ and its selfadjoint perturbations in the same Hilbert space $\mathcal{H}$. As S.Albeverio and P. Kurosov (2000), we call a selfadjoint operator $A_{1}$ the singular perturbation of $A$ if $A_{1}$ and {A} have different domains $\mathcal{D}(A),\mathcal{D}(A_{1})$ but $A=A_{1}$ on $\mathcal{D}(A)\cap\mathcal{D}(A_{1})$. Assuming that $A$ has absolutely continuous spectrum and the difference of resolvents $R_{z}(A_{1}) -R_{z}(A)$ of $A_{1}$ and $A$ for non-real $z$ is a trace class operator we find the explicit expression for the sc…
In the present paper we investigate the set $Σ_J$ of all $J$-self-adjoint extensions of a symmetric operator $S$ with deficiency indices $<2,2>$ which commutes with a non-trivial fundamental symmetry $J$ of a Krein space $(\mathfrak{H}, [\cdot,\cdot])$, SJ=JS. Our aim is to describe different types of $J$-self-adjoint extensions of $S$. One of our main results is the equivalence between the presence of $J$-self-adjoint extensions of $S$ with empty resolvent set and the commutation of $S$ with a Clifford algebra ${\mathcal C}l_2(J,R)$, where $R$ is an additional fundamental symmetry with $JR=-R…
A spectral theory of linear operators on rigged Hilbert spaces (Gelfand triplets) is developed under the assumptions that a linear operator $T$ on a Hilbert space $\mathcal{H}$ is a perturbation of a selfadjoint operator, and the spectral measure of the selfadjoint operator has an analytic continuation near the real axis in some sense. It is shown that there exists a dense subspace $X$ of $\mathcal{H}$ such that the resolvent $(λ-T)^{-1}ϕ$ of the operator $T$ has an analytic continuation from the lower half plane to the upper half plane as an $X'$-valued holomorphic function for any $ϕ\in X$, …
Given a pair of self-adjoint operators $H$ and $V$ such that $V$ is bounded and $(H+V-i)^{-1}-(H-i)^{-1}$ belongs to the Schatten-von Neumann ideal $\mathcal{S}^n$, $n\ge 2$, of operators on a separable Hilbert space, we establish higher order trace formulas for a broad set of functions $f$ containing several major classes of test functions and also establish existence of the respective locally integrable real-valued spectral shift functions determined uniquely up to a low degree polynomial summand. Our result generalizes the result of \cite{PSS13} for Schatten-von Neumman perturbations $V$ an…
We study the model operator $\mathbf{D}_{\mathbf{A}} = (d/dt) + \mathbf{A}$ in $L^2(\mathbb{R};\mathcal{H})$ associated with the operator path $\{A(t)\}_{t=-\infty}^{\infty}$, where $(\mathbf{A} f)(t) = A(t) f(t)$ for a.e.\ $t\in\mathbb{R}$, and appropriate $f \in L^2(\mathbb{R};\mathcal{H})$ (with $\mathcal{H}$ a separable, complex Hilbert space). Denoting by $A_{\pm}$ the norm resolvent limits of $A(t)$ as $t \to \pm \infty$, our setup permits $A(t)$ in $\mathcal{H}$ to be an unbounded, relatively trace class perturbation of the unbounded self-adjoint operator $A_-$, and no discrete spectrum…
Using the Davies-Helffer-Sjöstrand functional calculus based on almost analytic extensions, we address the following problem: Given self-adjoint operators $S_j$, $j=1,2$, in $\mathcal{H}$, and functions $f$ in an appropriate class, for instance, $f \in C_0^{\infty}(\mathbb{R})$, how to control the norm $\|f(S_2) - f(S_1)\|_{\mathcal{B}(\mathcal{H})}$ in terms of the norm of the difference of resolvents, $\|(S_2 - z_0 I_{\mathcal{H}})^{-1} - (S_2 - z_0 I_{\mathcal{H}})^{-1}\|_{\mathcal{B}(\mathcal{H})}$, for some $z_0 \in \mathbb{C}\backslash\mathbb{R}$. We are particularly interested in the ca…
Let $Ω_-$ and $Ω_+$ be two bounded smooth domains in $\mathbb{R}^n$, $n\ge 2$, separated by a hypersurface $Σ$. For $μ>0$, consider the function $h_μ=1_{Ω_-}-μ1_{Ω_+}$. We discuss self-adjoint realizations of the operator $L_μ=-\nabla\cdot h_μ\nabla$ in $L^2(Ω_-\cupΩ_+)$ with the Dirichlet condition at the exterior boundary. We show that $L_μ$ is always essentially self-adjoint on the natural domain (corresponding to transmission-type boundary conditions at the interface $Σ$) and study some properties of its unique self-adjoint extension $\mathcal{L}_μ:=\overline{L_μ}$. If $μ\ne 1$, then $\mat…
We consider the damped Schrödinger semigroup $e^{-it \frac{d^2}{dx^2}}$ on the tadpole graph ${\mathcal R}$. We first give a careful spectral analysis and an appropriate decomposition of the kernel of the resolvent. As a consequence and by showing that the generalized eigenfunctions form a Riesz basis of some subspace of $L^2({\mathcal R})$, we prove that the corresponding energy decay exponentially.
