Download Hilbert Space Operators in Quantum Physics by Jirí Blank, Pavel Exner, Miloslav Havlícek PDF

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We denote by X the set of all sequences x := {xj } , xj ∈ Xj , such that j xj j < ∞ , and equip it with the “componentwise” defined summation and scalar multiplication. The norm X ⊕ := j xj j turns it into a Banach space; the completeness can be checked as for p (Problem 23). The space (X , · ⊕ ) is ⊕ called the direct sum of the spaces Xj , j = 1, 2, . , and denoted as j Xj . 5 Banach spaces and operators on them 17 (ii) Starting from the same family {Xj : j = 1, 2, . }, one can define another Banach space (which is sometimes also referred to as a direct sum) if we change the above norm to X ∞ := supj xj j replacing, of course, X by the set of sequences for which X ∞ < ∞.

The dual space of a given V is normed, so we can define the second dual V ∗∗ := (V ∗ )∗ as well as higher dual spaces. For any x ∈ V we can define Jx ∈ V ∗ by Jx (f ) := f (x). The map x → Jx is a linear isometry of V to a subspace of V ∗∗ (Problem 55); if its range is the whole V ∗∗ the space V is called reflexive. It follows from the definition that any reflexive space is automatically Banach. 9, the spaces p are reflexive for p > 1, and the same is true for Lp (M, dµ) (see the notes). On the other hand, 1 and C(K) are not reflexive, and similarly, L1 (M, dµ) is not reflexive unless the measure µ has a finite support.

Any bounded operator is obviously closed. On the other hand, there are closed operators which are not bounded (Problem 59). If Γ(T ) is not closed but its closure in X ⊕ Y is a graph (which may not be true) the operator T is said to be closable and the closed operator T := TΓ(T ) is called the closure of the operator T ; we have Γ(T ) = Γ(T ). Since Γ(T ) is the smallest closed set containing Γ(T ), the closure is the smallest closed extension of the operator T . Moreover, Γ(T ) is a subspace in X ⊕ Y, so it is a graph iff [0, y] ∈ Γ(T ) implies y = 0.