Editing Isospectral (section)

{{Short description|Linear operators with a common spectrum}}
In [[mathematics]], two [[linear operator]]s are called '''isospectral''' or '''cospectral''' if they have the same [[spectrum of an operator|spectrum]]. Roughly speaking, they are supposed to have the same [[Set (mathematics)|sets]] of [[eigenvalue]]s, when those are counted with [[Multiplicity (mathematics)|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 [[matrix (mathematics)|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''&nbsp;=&nbsp;0.  The most studied isospectral problem in infinite dimensions is that of the [[Laplace operator]] on a domain in '''R'''<sup>2</sup>.  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]].