# What is the difference between Hermitian and self-adjoint?

What is the difference between Hermitian and self-adjoint? If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator A is self-adjoint if and only if the matrix describing A

## What is the difference between Hermitian and self-adjoint?

If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator A is self-adjoint if and only if the matrix describing A with respect to this basis is Hermitian, i.e. if it is equal to its own conjugate transpose. …

## What is the meaning of self-adjoint?

From Wikipedia, the free encyclopedia. In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space V with inner product. (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map A (from V to itself) that is its own adjoint.

## Are all self-adjoint operators Hermitian?

A is symmetric (or formally self-adjoint, apparently physicists call them also Hermitian, but no mathematician would do this) if A⊂A∗; self-adjoint if A=A∗. Thus every self-adjoint operator is symmetric, but the converse need not hold. However, if A is continuous and D(A)=H then A symmetric implies A self-adjoint.

## Are all positive operators self-adjoint?

Every positive operator A on a Hilbert space is self-adjoint.

## Are all normal operators self-adjoint?

(a) Every self-adjoint operator is normal. True: The formula to be normal (TT∗ = T∗T) is true when T = T∗. True: The (real) spectral theorem says that an operator is self-adjoint if and only if it has an orthonormal basis of eigenvectors. The eigenvectors given form an orthonormal basis for R2.

In particular, the sum of an unbounded self-adjoint operator and a bounded self-adjoint operator (defined on all of H) is self-adjoint on the domain of the unbounded operator. Proof. See Exercise 3. The sum of two unbounded self-adjoint operators is not, in general, self- adjoint.

## Is the zero operator self-adjoint?

While all orthogonal projections are self-adjoint, they are not unitary except for the trivial cases of the identity operator I and the zero operator 0. Proposition 1.7. The space of all self-adjoint operators on a Hilbert space H is closed in BL(H, H).