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Compact operators on Hilbert space
Published in Orr Moshe Shalit, A First Course in Functional Analysis, 2017
In Exercise 8.5.11 we saw how to “evaluate a function at an operator”, that is, how to make sense of g(A) when g is an analytic function on a disc Dr(0), with r > ║A║. The term functional calculus refers to the procedure of evaluating a function g (from within a specified class of functions) at an operator A (from within a specified class of operators), to form a new operator g(A). This notion also goes under the names function calculus, operational calculus, or function of an operator. The purpose of this section is to introduce the functional calculus for compact normal operators.
On the existence and regularity of admissibly inertial manifolds with sectorial operators
Published in Dynamical Systems, 2022
Thieu Huy Nguyen, Xuan-Quang Bui
The purpose of the present paper is to extend the result of Nguyen [30] to a more general case of the linear operator. We will prove the existence of an admissibly inertial manifold for the semilinear nonautonomous evolution equation (1) when is a sectorial operator. In our problem, the linear operator is just a sectorial operator in a general Banach space, and it is no longer self-adjoint nor has a discrete spectrum with a compact resolvent acting on a Hilbert space. This setting leads to an essential difficulty when working with the proof of the main result, that is, the spectral projection is no longer an orthogonal projection, so dichotomy estimates need to be modified to make it available in this situation. To overcome that difficulty, we suppose that the spectrum of is separated into two parts such that there is a sufficiently large distance between these two separated parts. This assumption allows us to use the Riesz projection, Dunford functional calculus and analytic semigroups to establish the dichotomy estimates for the semigroup generated by sectorial operator (see Proposition 2.2). These dichotomy estimates, together with the admissibility of function spaces representing the φ-Lipschitz property of the nonlinear term f, help to implement some procedures in the functional analysis combined with the use of Lyapunov–Perron equations to obtain the existence of an admissibly inertial manifold.
The coupled-cluster formalism – a mathematical perspective
Published in Molecular Physics, 2019
A. Laestadius, F. M. Faulstich
A key element in deriving the exponential parameterisation from the mathematical viewpoint is the well-definedness of the exponential of (or equivalently the logarithm of ), which is subject of functional calculus. We emphasise that the applicability of functional calculus depends strongly on the operator's domain since different domains may imply different properties of the operator, e.g. boundedness, essential self-adjointness, sectorial spectrum, etc. By the fact that Rohwedder [16] showed the -continuity of cluster operators in a continuous setting, the functional calculus for bounded operators was proven to be applicable.