NOTATION GUID E fJOO ^on i, -invariant HQQL -simply invarian t H^L-M\ rang e Block invariant •-Block invarian t EJOO N{u) n(£) n L (n) H 0UL(u) M±J [F,G]j n? A[a) The set of functions L i n H™xn such that 7 5 L(0) G OL , th e algebra of lower triangular n x n matrices. Subspaces M o f L\xn suc h that F E M , 7 6 L 6 i ^ L = * FL M- Invariant subspace M suc h that 7 6 n L e H S . n [ {n:feM} = (0). Invariant subspace M suc h that 7 6 UL€L°° {FL: F G M} is dense in L\ xn . A subspace which is #onL -simply invariant 7 6 and HoQ L -fu\l range. Subspace which is simply invariant an d full 7 6 range with respect to #on u {F Lfxn: F E H™xn and F(0) belon g to 7 6 Qu th e algebra o f n xn uppe r triangula r matrices}. F =comple x conjugat e (pointwise and elementwise) o f F. { I / I , . . . , I / } C N » . 7 6 ^i + -- « + ^ n . 7 6 The set of all N(v) x n matrices, written a s 7 6 n x n block matrices F = [Fij]iij n , where the size of (i, j) bloc k Fij i s i/« x 1. Set of all block matrices F = [Ftj]it,j n 7 6 which are block lower triangular (Fij = 0 for i j). Set of all block upper triangular matrices. 7 6 {Fetf£ (K)xn :F(o)en L ( £ )}. 7 6 Orthogonal complement o f M i n the 7 7 J-inner product . ^ J 0 2n tr(G*(e i0 )JF(ei0)) d$, defined o n 7 7 Lfcxn' {[Fij] E QL: F^ = 0 for t j + m}. 8 1 Chapter 8 The intersection 8 9 Sf) Ay i f ! Ay2 n-'-f l AyL . Chapter 9 Principal submatrix of^4 contained i n the 9 5 rows and columns indicated by the index set a,/? C {1,2,...,n} .
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