Between the late 1940s and early 1960s the techniques, previously developed as part of classical potential theory, were abstracted within operator theory by various mathematicians, including M. G. Krein, William T. Reid, Peter Lax and Jean Dieudonné.
Finally the eigenspaces of K* span a dense subspace of H, since it contains the image under R of the corresponding space for A.
The above arguments also imply that the eigenvectors for non-zero eigenvalues of KS in HS all lie in the subspace H. Hilbert–Schmidt operators K with non-zero real eigenvalues λn satisfy the following identities proved by Carleman (1921): Here tr is the trace on trace-class operators and det is the Fredholm determinant.
For symmetrizable operators, the identities for K* can be proved by taking H0 to be the kernel of K* and Hm the finite dimensional eigenspaces for the non-zero eigenvalues λm.
Let PN be the orthogonal projection onto the direct sum of Hm with 0 ≤ m ≤ N. This subspace is left invariant by K*.