What is "isospectral"

a. (context mathematics English) Having the same spectrum

In mathematics, two linear operators are called isospectral or cospectral if they have the same spectrum. Roughly speaking, they are supposed to have the same sets of eigenvalues, when those are counted with multiplicity.

The theory of isospectral operators is markedly different depending on whether the space is finite or infinite dimensional. In finite-dimensions, one essentially deals with square matrices.

In infinite dimensions, the spectrum need not consist solely of isolated eigenvalues. However, the case of a compact operator on a Hilbert space (or Banach space) is still tractable, since the eigenvalues are at most countable with at most a single limit point λ = 0. The most studied isospectral problem in infinite dimensions is that of the Laplace operator on a domain in R. Two such domains are called isospectral if their Laplacians are isospectral. The problem of inferring the geometrical properties of a domain from the spectrum of its Laplacian is often known as hearing the shape of a drum.