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Register a new userOn self-adjoint operators in Krein spaces constructed by Clifford algebra Cl_2
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Let $J$ and $R$ be anti-commuting fundamental symmetries in a Hilbert space $mathfrak{H}$. The operators $J$ and $R$ can be interpreted as basis (generating) elements of the complex Clifford algebra ${mathcal C}l_2(J,R):={span}{I, J, R, iJR}$. An arbitrary non-trivial fundamental symmetry from ${mathcal C}l_2(J,R)$ is determined by the formula $J_{vec{alpha}}=alpha_{1}J+alpha_{2}R+alpha_{3}iJR$, where ${vec{alpha}}inmathbb{S}^2$. Let $S$ be a symmetric operator that commutes with ${mathcal C}l_2(J,R)$. The purpose of this paper is to study the sets $Sigma_{{J_{vec{alpha}}}}$ ($forall{vec{alpha}}inmathbb{S}^2$) of self-adjoint extensions of $S$ in Krein spaces generated by fundamental symmetries ${{J_{vec{alpha}}}}$ (${{J_{vec{alpha}}}}$-self-adjoint extensions). We show that the sets $Sigma_{{J_{vec{alpha}}}}$ and $Sigma_{{J_{vec{beta}}}}$ are unitarily equivalent for different ${vec{alpha}}, {vec{beta}}inmathbb{S}^2$ and describe in detail the structure of operators $AinSigma_{{J_{vec{alpha}}}}$ with empty resolvent set.
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Let $X$ be a space of homogeneous type and let $L$ be a nonnegative self-adjoint operator on $L^2(X)$ which satisfies a Gaussian estimate on its heat kernel. In this paper we prove a Homander type spectral multiplier theorem for $L$ on the Besov and Triebel--Lizorkin spaces associated to $L$. Our work not only recovers the boundedness of the spectral multipliers on $L^p$ spaces and Hardy spaces associated to $L$, but also is the first one which proves the boundedness of a general spectral theorem on Besov and Triebel--Lizorkin spaces.
We compute the deficiency spaces of operators of the form $H_A{hat{otimes}} I + I{hat{otimes}} H_B$, for symmetric $H_A$ and self-adjoint $H_B$. This enables us to construct self-adjoint extensions (if they exist) by means of von Neumanns theory. The structure of the deficiency spaces for this case was asserted already by Ibort, Marmo and Perez-Pardo, but only proven under the restriction of $H_B$ having discrete, non-degenerate spectrum.
The paper is devoted to a development of the theory of self-adjoint operators in Krein spaces (J-self-adjoint operators) involving some additional properties arising from the existence of C-symmetries. The main attention is paid to the recent notion of stable C-symmetry for J-self-adjoint extensions of a symmetric operator S. The general results are specialized further by studying in detail the case where S has defect numbers <2,2>.
Various norms can be defined on a Krein space by choosing different underlying fundamental decompositions. Some estimates of norms on Krein spaces are discussed and few results in Bognars paper are generalized.
We extend the classical boundary values begin{align*} & g(a) = - W(u_{a}(lambda_0,.), g)(a) = lim_{x downarrow a} frac{g(x)}{hat u_{a}(lambda_0,x)}, &g^{[1]}(a) = (p g)(a) = W(hat u_{a}(lambda_0,.), g)(a) = lim_{x downarrow a} frac{g(x) - g(a) hat u_{a}(lambda_0,x)}{u_{a}(lambda_0,x)} end{align*} for regular Sturm-Liouville operators associated with differential expressions of the type $tau = r(x)^{-1}[-(d/dx)p(x)(d/dx) + q(x)]$ for a.e. $xin[a,b] subset mathbb{R}$, to the case where $tau$ is singular on $(a,b) subseteq mathbb{R}$ and the associated minimal operator $T_{min}$ is bounded from below. Here $u_a(lambda_0, cdot)$ and $hat u_a(lambda_0, cdot)$ denote suitably normalized principal and nonprincipal solutions of $tau u = lambda_0 u$ for appropriate $lambda_0 in mathbb{R}$, respectively. We briefly discuss the singular Weyl-Titchmarsh-Kodaira $m$-function and finally illustrate the theory in some detail with the examples of the Bessel, Legendre, and Kummer (resp., Laguerre) operators.
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