China
Africa
Explore chapters and articles related to this topic
Hilbert spaces
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
Theorem 10.2 (Spectral theorem for compact self-adjoint operators) Let H be a Hilbert space, and let T : H → H be a compact, self-adjoint operator. Then:There is a complete orthonormal system {ϕn} in H consisting of eigenvectors of T.The eigenvalues are real (by self-adjointness).Each eigenspace is finite-dimensional (by compactness).The only possible limit point for a sequence of these eigenvalues is 0 (by compactness).||T|| = |λ|, where λ is the eigenvalue of largest magnitude.
Linear Algebra
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
One of the most important theorems in linear algebra is the Spectral Theorem. This theorem tells us when a matrix can be diagonalized. In fact, it goes beyond matrices to the diagonalization of linear operators. We learn in linear algebra that linear operators can be represented by matrices once we pick a particular representation basis. Diagonalization is simplest for finite dimensional vector spaces and requires some generalization for infinite dimensional vectors spaces. Examples of operators to which the spectral theorem applies are self-adjoint operators (more generally normal operators on Hilbert spaces). We will explore some of these ideas later in the course. The spectral theorem provides a canonical decomposition, called the spectral decomposition, or eigendecomposition, of the underlying vector space on which it acts.
Compact operators on Hilbert space
Published in Orr Moshe Shalit, A First Course in Functional Analysis, 2017
In the previous chapter, we introduced and studied compact operators on Banach spaces. In this chapter, we will restrict attention to compact operators on Hilbert spaces, where, due to the unique geometry of Hilbert spaces, we will be able to say much more. The culmination of this chapter is the so-called spectral theorem for compact normal operators on a Hilbert space. The spectral theorem is the infinite dimensional version of the familiar fact that every normal matrix is unitarily diagonalizable.
Regularization of linear ill-posed problems involving multiplication operators
Published in Applicable Analysis, 2022
P. Mathé, M. T Nair, B. Hofmann
As outlined in the introduction the setup of multiplication operators as analyzed here is prototypical for general bounded self-adjoint positive operators mapping in the (separable) Hilbert space H due to the associated Spectral Theorem (cf. [1]), stated as Fact. It is an advantage of our focus on multiplication operators that we can include compact linear operators and non-compact ones as well.