By Barbatis G., Davies E.B., Erdos J.A.
Those lecture notes are in line with the classes "Operator conception" constructed atKing's collage London by means of G. Barbatis, E.B. Davies and J.A. Erdos, and"Functional research II" built on the collage of Sussex by way of P.J. Bushell,D.E. Edmunds and D.G. Vassiliev. As traditional, all mistakes are completely my accountability. an analogous applies to the accompanying workout sheets.
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Part (v) shows that it is, in fact, a C ∗ –algebra. A mapping x → x∗ of an algebra E into itself is called an involution if it has the following properties (αx + βy)∗ = αx∗ + βy ∗ , (xy)∗ = y ∗ x∗ , x∗∗ = x, for all x, y ∈ E and α, β ∈ IF. A Banach algebra E with an involution satisfying the identity x∗ x = x 2 , ∀x ∈ E, is called a C ∗ –algebra. 3(i) for operators acting on normed spaces has the following form (αB1 + βB2 )∗ = αB1∗ + βB2∗ , ∀α, β ∈ IF. 3(i). 2 and that for operators acting on normed spaces.
So, f is a bounded linear functional on H1 and f ≤ B y . 34) there exists a unique z = z(B, y) ∈ H1 such that (Bx, y) = f (x) = (x, z), ∀x ∈ H1 , and z = f ≤ B y . ). e. B ∗ is bounded and B∗ ≤ B . 1) is satisfied and that the constructed operator B ∗ is the unique operator satisfying this equality. 2. Definition the adjoint of B. 3. Theorem Let H1 , H2 and H3 be Hilbert spaces, B, B1 , B2 ∈ B(H1 , H2 ) and T ∈ B(H2 , H3 ). Then (i) (αB1 + βB2 )∗ = αB1∗ + βB2∗ , ∀α, β ∈ IF; (ii) (T B)∗ = B ∗ T ∗ ; (iii) B ∗∗ = B; (iv) B ∗ = B ; (v) B ∗ B = BB ∗ = B 2 ; (vi) if B is invertible then so is B ∗ and (B ∗ )−1 = (B −1 )∗ .
We have N lin{en }N n=1 = lin(eN , LN −1 ) = lin(uN , LN −1 ) = lin(zN , LN −1 ) = lin{zn }n=1 . 22. Consequently eN is orthogon=1 −1 nal to any element of the orthonormal set {en }N n=1 . It is clear that eN = 1. So the set {en }N n=1 is orthonormal. This completes the induction step. Thus we have constructed an orthonormal set {en } such that lin{en } = lin{zn } = lin{yn }. 30. Definition A normed space is called separable if it contains a countable dense subset. 31. Theorem A Hilbert space H is separable iff it contains a countable (or finite) complete orthonormal set.
Operator theory by Barbatis G., Davies E.B., Erdos J.A.
by David
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