I: Functional Analysis, Volume 1 (Methods of Modern by Michael Reed

By Michael Reed

This e-book is the 1st of a multivolume sequence dedicated to an exposition of useful research tools in smooth mathematical physics. It describes the elemental ideas of useful research and is largely self-contained, even supposing there are occasional references to later volumes. now we have incorporated a number of purposes after we notion that they'd supply motivation for the reader. Later volumes describe a number of complicated subject matters in sensible research and provides various functions in classical physics, smooth physics, and partial differential equations.

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I: Functional Analysis, Volume 1 (Methods of Modern Mathematical Physics) (vol 1)

This ebook is the 1st of a multivolume sequence dedicated to an exposition of practical research tools in smooth mathematical physics. It describes the elemental ideas of sensible research and is basically self-contained, even though there are occasional references to later volumes. we've got integrated a number of purposes after we concept that they might supply motivation for the reader.

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Example text

Then c 2 f (c) = (ηf )2 (c) − (ηf )2 (a) = a ≤2 f f + 2M f ≤ ε2 f 2 2 c (ηf )2 (t) dt = = 2ε f ε −1 f + 2M f + ε −2 + 2M f 2 2ηf ηf +η f dt a ≤ ε f 2 + ε −1 + 1 + M f 2 . d on the half-axis and on the Next we investigate the differentiation operator −i dx whole axis. For J = (0, ∞) or J = R, we define H 1 (J ) = f ∈ L2 (J ) : f ∈ AC[a, b] for all [a, b] ⊆ J and f ∈ L2 (J ) . Let J = (0, +∞). 11(i), we obtain for f, g ∈ H 1 (0, +∞), f , g + f, g = −f (0)g(0). 6 (Half-axis) Let T be the operator on L2 (0, +∞) defined by Tf = −if for f in the domain D(T ) = H01 (0, ∞) := {f ∈ H 1 (0, +∞) : f (0) = 0}.

Since D(S) = H1 by assumption and D(S ∗ ) = H1 because S is bounded, we have D(T + S) = D(T ) and D(T ∗ +S ∗ ) = D(T ∗ ). Let y ∈ D((T +S)∗ ). For x ∈ D(T +S), T x, y = (T + S)x, y − Sx, y = x, (T + S)∗ y − S ∗ y . This implies that y ∈ D(T ∗ ) = D(T ∗ + S ∗ ). Let T be a densely defined closable linear operator from H1 into H2 . 8(ii) below. 6(ii), applied to T and T ∗ , yields N T ∗ = R(T )⊥ , H2 = N T ∗ ⊕ R(T ), N (T ) = R T ∗ ⊥ , H1 = N (T ) ⊕ R T ∗ . 6) Note that the ranges R(T ) and R(T ∗ ) are not closed in general.

Find examples of densely defined closed operators T and S on a Hilbert space for which (T + S)∗ = T ∗ + S ∗ . 9. Let T be a closed operator on H, and let B ∈ B(H). Show that the operators T B and T + B are closed. 10. Find a closed operator T on H and an operator B ∈ B(H) such that BT is not closed. Hint: Take B = ·, x y, where x ∈ / D(T ∗ ) and y = 0. 11. Let T denote the multiplication operator by ϕ ∈ C(R) on L2 (R) with domain D(T ) = {f ∈ L2 (R) : ϕ · f ∈ L2 (R)}. Show that C0∞ (R) is a core for T .

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