The Naimark Representation
Arthur E. Frazho, Wisuwat Bhosri · Birkhäuser Basel eBooks · 2009
This chapter is devoted to the Naimark representation theorem and its consequences. The Naimark dilation allows us to use geometric methods to compute inner-outer factorizations and solve signal processing problems. Let A be any matrix whose entries A jk are operators mapping a Hilbert space E into Y. Then A# denotes the matrix obtained by taking the adjoint of the entries of A and then transposing this matrix, that is, the entries of A# are given by (A#) jk =A* kj . If A defines an operator mapping ⊕ 0 n ε into ⊕ 0 m Y, then A#=A* is the adjoint of A. Throughout l+ c (ε) denotes the set of all vectors in l+2(ε) with compact support. Finally, recall that the controllability matrix W determined by the pair of operators {A on χ,B where B maps ε into χ is given by (5.0.1) $$ W = [B AB A^2 B \cdots ].$$ In general, W is not necessarily an operator mapping l+2(ε) into ⊕. However, W is a well-defined linear map from l+ c (ε) into ⊕.