Abstract
In recent ten years, there has been much concentration and increased research activities on Hamilton’s Ricci flow evolving on a Riemannian metric and Perelman’s functional. In this paper, we extend Perelman’s functional approach to include logarithmic curvature corrections induced by quantum effects. Many interesting consequences are revealed.
Highlights
The present paper is devoted to the spectrum localization of regular nonselfadjoint differential operators whose coefficients are bounded operators acting in a separable Hilbert space, and the norm estimates for their resolvents.The literature on the theory of abstract differential operators is rather rich, but mainly is devoted to the coercitivity of operators and maximal regularity of solutions of the relevant equations, cf. the well-known books [11, 12, 14] and references therein
The monograph [13] considers the interplay between spectral and oscillatory properties of both finite and infinite systems of linear ordinary differential selfadjoint operators. These can be written as single differential equations with matrix-valued and operator-valued coefficients, respectively
Note that inequality (8.3) is a particular case of the well-known Theorem V.5.1 from [9], but we suggest a considerably new approach
Summary
Resolvent and spectrum of a nonselfadjoint differential operator in a Hilbert space Abstract. We consider a second order regular differential operator whose coefficients are nonselfadjoint bounded operators acting in a Hilbert space. An estimate for the resolvent and a bound for the spectrum are established. An operator is said to be stable if its spectrum lies in the right half-plane.
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