Operator Matrices

Abstract

An orthonormal basis serves to express a Hilbert space as the direct sum of one-dimensional subspaces. Some of the matrix theory associated with orthonormal bases deserves to be extended to more general direct sums. Suppose, to be specific, that H = H ₁ ⊕ H ₂ ⊕ H ₃ ⊕ ⋯. (Uncountable direct sums work just as well, and finite ones even better.) If the direct sum is viewed as an “internal” one, so that the H _i’s are subspaces of H, then the elements f of H are sums

$$ f = {f_{{1\,}}} + {f_2} + {f_3} + = \cdots, $$

with f _i in H _i. If A is an operator on H, then

$$ Af = A\,{f_{{1\,}}} + A{f_2} + A{f_3} + = \cdots . $$

Each Af _j , being an element of H, has a decomposition:

$$ A{f_j} = {g_{{1j}}} + {g_{{2j}}} + {g_{{3j}}} + \cdots, $$

with g _ij in H _i. The g _ij’s depend, of course, on f _j , and the dependence is linear and continuous. It follows that

$$ {g_{{ij\,}}} = {A_{{ij}}}\,{f_i} $$

where A _ij is a bounded linear transformation from H _j to H _i . The construction is finished: corresponding to each A on H there is a matrix <A _ij >, whose entry in row; and column j is the projection onto the i component of the restriction of A to H _j .