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Chapter 13: Problem 32

Show that if \(\left(A, D_{A}\right)\) and \(\left(B, D_{B}\right)\) arc operators on dense domains in \(H\) then \(B^{*} A^{*} \subseteq\) \((A B)^{\circ}\)

Short Answer

Expert verified
The proof requires understanding of adjoint operators in a Hilbert space. Given \(x \in B^{*} A^{*}\), demonstrates that \(x\) is also in \((A B)^{\circ}\), thereby showing that \(B^{*} A^{*} \subseteq (A B)^{\circ}\).

Step by step solution

01

Understanding the problem

You should understand the operation of adjoint and closure.
02

Assume x is in \(B^{*} A^{*}\)

Suppose an element \(x\) belongs to \(B^{*} A^{*}\). That means there exists \(y\) in \(D_{A}\) and \(z\) in \(D_{B}\) such that \(x= B^{*} A^{*} y = B^{*} z\).
03

Show x is in \((A B)^{\circ}\)

To prove that \(B^{*} A^{*} \subseteq (A B)^{\circ}\), you need to show that \(x\) is also in \((A B)^{\circ}\), that is, \(x=(A B)^{\circ} y\). Using definition of closure, we know \((A B)^{\circ}\) is the smallest closed set that contains \(A B\). If \(x\in B^{*} A^{*}\), then \(x\) will also belongs to the closure of \(A B\), \((A B)^{\circ}\).

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Most popular questions from this chapter

In the Hulbert space \(L^{2}([-1,1])\) let \(\left.\mid f_{n}(x)\right\\}\) be the sequence of functions \(1, x, x^{2}, \ldots, f_{n}(x)=x^{n} \ldots .\) (a) Apply Schmidt orthonormalization to this sequence, wnting down the first three polynomials so obtained. (b) The \(n\)th Legendre polynomial \(P_{n}(x)\) is defined as $$ P_{n}(x)=\frac{1}{2^{n} n !} \frac{d^{n}}{\mathrm{dx}^{n}}\left(x^{2}-1\right)^{n} $$ Prove that $$ \int_{-1}^{1} P_{m}(\mathrm{r}) P_{n}(\mathrm{x}) \mathrm{d} \mathrm{x}=\frac{2}{2 n+1} \delta_{m \mathrm{~m}} $$ (c) Show that the \(n\)th member of the o.n. sequence obtained in (a) is \(\sqrt{n+\frac{1}{2}} P_{n}(x)\).

If \(\left(A, D_{A}\right)\) is a densely defined openator and \(D_{A}\) is dense in \(\mathcal{H}_{1}\) show that \(A \subseteq A^{* *}\).

For every bounded operator \(A\) on a Hilbert space \(\mathcal{H}\) show that the exponential operator $$ \mathrm{e}^{A}=\sum_{n=0}^{\infty} \frac{A^{n}}{n !} $$ is well-defined and bounded on \(\mathcal{H}\). Show that (a) \(e^{0}=1\) (b) For all posituve integers \(n,\left(c^{A}\right)^{n}=e^{n A}\). (c) \(\mathrm{e}^{A}\) is invertuble for all bounded operators \(A\) (even if \(A\) is not mivertible) and \(e^{-A}=\left(e^{4}\right)^{-1} .\) (d) If \(A\) and \(B\) are commuting operators then \(e^{A+B}=\mathrm{e}^{A} \mathrm{e}^{\theta}\) (c) If \(A\) is hermitian then \(e^{i A}\) is unitary.

Let \(E\) be a Banach space in which the norm satisfies the parallelogram law (13.2). Show that it is a Hilbert space with inner product given by $$ \langle x \mid y\rangle=\frac{1}{4}\left(\|x+y\|^{2}-\|x-y\|^{2}+\imath\|x-i y\|^{2}-i \| x+\left.1 y\right|^{2}\right) $$

Let \(A\) be a bounded openitor on a Hilbert space \(\mathcal{H}\) with a one- damensional rangc. (a) Show that there exist vectors \(u, v\) such that \(A x=\langle v \mid x\rangle u\) for all \(x \in \mathcal{H}\). (b) Show that \(A^{2}=\lambda A\) for some scalar \(\lambda\), and that \(\|A\|=\|u\|\|v\|\). (c) Prove that \(A\) is hermitian, \(A^{*}=A\), If and only if there exists a real number \(a\) such that \(v=a u\).
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