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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.
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
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
Let $L$ be a non-negative self-adjoint operator acting on the space $L^2(X)$, where $X$ is a metric measure space. Let ${ L}=int_0^{infty} lambda dE_{ L}({lambda})$ be the spectral resolution of ${ L}$ and $S_R({ L})f=int_0^R dE_{ L}(lambda) f$ denot
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 arb
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