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Register a new userOn J-Self-Adjoint Operators with Stable C-Symmetry
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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>.
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A well known tool in conventional (von Neumann) quantum mechanics is the self-adjoint extension technique for symmetric operators. It is used, e.g., for the construction of Dirac-Hermitian Hamiltonians with point-interaction potentials. Here we reshape this technique to allow for the construction of pseudo-Hermitian ($J$-self-adjoint) Hamiltonians with complex point-interactions. We demonstrate that the resulting Hamiltonians are bijectively related with so called hypermaximal neutral subspaces of the defect Krein space of the symmetric operator. This symmetric operator is allowed to have arbitrary but equal deficiency indices $<n,n>$. General properties of the $cC$ operators for these Hamiltonians are derived. A detailed study of $cC$-operator parametrizations and Krein type resolvent formulas is provided for $J$-self-adjoint extensions of symmetric operators with deficiency indices $<2,2>$. The technique is exemplified on 1D pseudo-Hermitian Schrodinger and Dirac Hamiltonians with complex point-interaction potentials.
In the present paper we investigate the set $Sigma_J$ 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 $JR=-RJ$. This enables one to construct the collection of operators $C_{chi,omega}$ realizing the property of stable $C$-symmetry for extensions $AinSigma_J$ directly in terms of ${mathcal C}l_2(J,R)$ and to parameterize the corresponding subset of extensions with stable $C$-symmetry. Such a situation occurs naturally in many applications, here we discuss the case of an indefinite Sturm-Liouville operator on the real line and a one dimensional Dirac operator with point interaction.
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.
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.
The main result (roughly) is that if (H_i) converges weakly to H and if also f(H_i) converges weakly to f(H), for a single strictly convex continuous function f, then (H_i) must converge strongly to H. One application is that if f(pr(H)) = pr(f(H)), where pr denotes compression to a closed subspace M, then M must be invariant for H. A consequence of this is the verification of a conjecture of Arveson, that Theorem 9.4 of [Arv] remains true in the infinite dimensional case. And there are two applications to operator algebras. If h and f(h) are both quasimultipliers, then h must be a multiplier. Also (still roughly stated) if h and f(h) are both in pA_sa p, for a closed projection p, then h must be strongly q-continuous on p.
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