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On the role of the comparison function in the spectral theory of selfadjoint operators

Commun. Math. Anal. Conf.
Conference 3 (2011), 77 - 87

On the role of the comparison function in the spectral theory of selfadjoint operators

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Abstract

Let A and B be selfadjoint lower-semibounded operators in L2(E;m). Assuming that the semigroups fe¡tA; t ¸ 0g and fe¡tB; t ¸ 0g are L2 ¡L¥ smoothing, the comparison function is given by Dt (x) = Z E je¡tB(x;y)¡e¡tA(x;y)j dm(y): We show that, provided the spectrum of A is known, integral conditions for Dt determine the essential spectrum, the absolutely continuous spectrum and the moments of the eigenvalues of B.