On the spectral theory of symmetric finite operators
Harro Heuser · Transactions of the American Mathematical Society · 1960
Let A be a linear operator defined on a linear system X and let N(A -XI) be the null space, R(A -XI) the range of A -XI, and X an arbitrary complex number. We call A a finite operator if for each X,z;0 the dimensions of N(A -XI) and X/R(A -XI) are finite and equal. The present paper is concerned with an iteration method for determining characteristic values and characteristic elements of symmetric finite operators on a not necessarily complete Hilbert space X and with the structure of the spectrum of such operators. The following two theorems are the basis of our exposition. THEOREM 1. If A is a symmetric finite operator on X and Ca(A) its continuous spectrum, then Ca(A) - { 0 } consists of all the limit points of characteristic values of A which are different from zero and no characteristic values themselves(2). THEOREM 2. If A is a symmetric finite operator on X and A 5O, then A has a characteristic value different from zero and each element Ax can be expanded in a series