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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 ⇒ Tx∉K. This generalizes subspaces invariant for both T and T−1 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.
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