Sselfadjoint

S-selfadjoint, often written S-selfadjoint, refers to operators on a Hilbert space that are self-adjoint with respect to a fixed positive, bounded, invertible operator S. An operator A is S-selfadjoint if it satisfies A* S = S A. Equivalently, A is self-adjoint in the inner product ⟨x,y⟩_S = ⟨Sx,y⟩, since the adjoint with respect to this inner product is S^{-1} A* S.

A key consequence is that, when S is positive, A is similar to a self-adjoint operator. Indeed, the operator B = S^{1/2} A S^{-1/2} is self-adjoint, and A = S^{-1/2} B S^{1/2}. Thus A has real spectrum, and many spectral properties of A can be studied through B. Eigenvectors correspond through the S-weighted change of basis, via x ↦ S^{1/2} x.

Special cases clarify the notion. If A is self-adjoint in the original inner product, A is S-selfadjoint precisely when A commutes with S (AS = SA). Conversely, S-selfadjointness does not imply A is self-adjoint unless S is a scalar multiple of the identity.

Applications of S-selfadjointness appear in operator theory and mathematical physics, particularly in the study of non-self-adjoint operators with real spectra, quasi-Hermitian operators, and pseudo-Hermitian quantum mechanics. The concept is also central in Krein space theory, where S defines a metric operator for an indefinite inner product.

Example: in finite dimensions, A = diag(2,3) is self-adjoint. With S = diag(1,2), we have A* S = S A, so A is S-selfadjoint, illustrating the general principle.