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By Cohn P.M.

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3 Other Topologies on L(H) L(H) has several other useful topologies in addition to the norm topology. 1 Definition. e. Ti → T strongly if Ti ξ → T ξ for all ξ ∈ H. e. Ti → T weakly if Ti ξ, η → T ξ, η for all ξ, η ∈ H. The word “operator” is often omitted in the names of these topologies; this usually causes no confusion, although technically there is a different topology on L(H) (more generally, on any Banach space) called the “weak topology,” the topology of pointwise convergence for linear functionals in L(H)∗ .

1 Definition. Let T be a densely defined operator on H. Then T is symmetric if T ⊆ T ∗ , and T is self-adjoint if T = T ∗ . A symmetric operator is closable (its closure is also symmetric) and a selfadjoint operator is closed. 2 Examples. 7 Unbounded Operators 31 D0 = {f ∈ D2 : f (0) = f (1) = 0}. Then D0 is dense in L2 [0, 1], and Dk has codimension one in Dk+1 (k = 0, 1). Let Tk be defined by Tk f = f with domain Dk (k = 0, 1, 2). Then each Tk is closed, T1 is self-adjoint, and T0∗ = T2 , so T0 is closed and symmetric, but not self-adjoint.

Tn and other entries 0. 4 ˜ T˜. Then W ˜ has first row W1 , . . , Wn and other entries 0. e. V ∗ V and V V ∗ are projections which are naturally the right and left support projections for T . This construction is independent of the choice of J (if K is another involution, then JK is unitary). We also have JT = |T ∗ J|U and (JT T ∗ J)n = J(T T ∗ )n J for all n, so |T ∗ J| = (JT T ∗ J)1/2 = J|T ∗ |J and hence T = J(JT ) = |T ∗ |JU = |T ∗ |V as in the linear case. Thus bounded conjugate-linear operators have a well-defined and unique polar decomposition with analogous properties to those of linear operators.

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