Vol. 68, No. 2, 1977

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Inner invariant subspaces

Gary S. Itzkowitz

Vol. 68 (1977), No. 2, 455–484

We single out a special subclass of the invariant subspaces which we call the inner invariant (i.i.) subspaces. A closed subspace K of a Hilbert space H is said to be i.i. for a linear operator T (with domain D) if: (1) T(K D) K, (2) {T(K D) + (K D)} = K, and (3) x D K TxK. This generalizes subspaces invariant for both T and T1 when the latter exists.

Some of the results in this paper are:

1. Let λ C. If |λ| < 1 then K is i.i. for U λ where U is the shift on Hardy space Hp iff K = gHp where g is inner and g(λ)0. If |λ|1, then K is i.i. for U λ iff K = gHp where g is inner. 2. There is an isometry J from H2 onto L2(0,) such that the i.i. subspaces of V + 1 (where V f(x) = 0Xf(y)dy) are precisely the subspaces J(gH2) for g an inner function. 3. Any skew-symmetric simple operator with defect indices (0,1) is isomorphic with V and V 1.

Mathematical Subject Classification 2000
Primary: 47A15
Received: 30 May 1975
Published: 1 February 1977
Gary S. Itzkowitz