For question involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.

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31 views

State of a $ C^{*} $-algebra.

Let $ (\pi,\mathcal{H}) $ be a non-degenerate $ * $-representation of a $ C^{*} $-algebra $ A $, and let $ h \in \mathcal{H} $ with $ \| h \| = 1 $. Define $ f_{h}: A \to \Bbb{C} $ by $ {f_{h}}(a) ...
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1answer
62 views
+50

When is the Laplace Beltrami Operator self-adjoint?

The Laplace-Beltrami operator is an operator which is the typical example of a self-adjoint operator in $L^{2}$. I am wondering if this is also true for other Hilbert spaces $W^{k,2}$. If this is ...
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0answers
7 views

unique inner product on a tensor product of Hilbert $C^*$ modules and Hilbert spaces.

For a $C^*-$ algebra $A$ and a Hilbert space $H$ and a Hilbert $A-$module E; how can we show that there is a unique $A-$ valued inner product on $H \otimes E$ as $< h_1 \otimes x_1 , h_2 \otimes ...
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1answer
51 views

Lemma 3.5-3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Is the set of non-zere Fourier co-efficients uncountable too?

Let $X$ be an inner product space, let $x \in X$ be non-zero, and let $M$ be an uncountable orthonormal subset of $X$. Then what can we say about the cardinality of the following set? $$ \{ \ v \in M ...
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10 views

$A$ be a non-empty closed convex subset of a Hilbert space $H$ , is the distance from $A$ always attained at a unique point in $A$ ?

Let $A$ be a non-empty closed convex subset of a Hilbert space $H$ , then is it true that for every $b \in H$ , $\exists$ unique $x_b \in A$ such that $||x_b-b||=d(b,A)=\inf \{||b-x||:x \in A\}$ ?
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2answers
22 views

Norm is sup of inner products (proof).

Let $V$ be a vector space with an inner product $\langle.,. \rangle$ and associated norm $|| . ||$ Then: Could I have a proof of this fact?
2
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1answer
318 views

Double orthogonal complement of any closed subspace is it self

Let $H$ be a pre-Hilbert space such that any closed sub space $M \subset H$ has the property $M^{\bot \bot}=M$. Prove that $H$ is a Hilbert space (ie, prove that $H$ is complete) My attempt: As ...
4
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1answer
43 views

Isolated Eigenvalue

What does it mean that an eigenvalue is "isolated"? My intuitive understanding says it is when one can find an open ball around it such that there is no other eigenvalue in that open ball. However, I ...
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1answer
22 views

On the proof of the continuity of the inner product.

I am having problems with the following proof and I need to fill in some details: I understand that continuity is being proven by the sequence definition but I do not get why (a) follows ...
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1answer
18 views

What can one assume about $T^*$ when showing that $T$ is normal?

Consider a continuous and linear operator $T$ such that $$ T : l^2 \to l^2 $$ where $(a_n) \mapsto (\alpha_n a_n)$ Moreover $(\alpha_n)$ is a sequence of complex numbers that converges to zero. Now, ...
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1answer
36 views

Normal Operators: Transform

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(T)\to\mathcal{H}:\quad N^*N=NN^*$$ Construct the operator: $$W:=(1+N^*N)^{-1}:\quad Z:=N\sqrt{W}$$ Then it is ...
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1answer
113 views

If $\sum (a_n)^2$ converges and $\sum (b_n)^2$ converges, does $\sum (a_n)(b_n)$ converge?

Could someone help me to solve this or at least give me a hint?, I have tried using Cauchy's criterion, the Dirichlet test for convergence, etc, but I can´t prove it.Honestly I don´t know where to ...
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0answers
18 views

Norm of operator matrix

I'm having trouble with the following: suppose H is a Hilbert space and $f_{i, j}, g_{i, j} : H \rightarrow H$, $1 \leq i, j \leq n$ are bounded operators. Then we have operators $(f_{i, j}) , (g_{i, ...
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1answer
32 views

Hilbert space L2 - inner product

I have a problem with one exercise. I have to prove that $L^2$ space is Hilbertian. So I think that the best way is to check out inner product by definition of norm, so: \begin{equation*} ...
2
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1answer
40 views

Showing that an operator generates a contraction semigroup

Let $A$ be the infinitesimal generator of a contraction semigroup $(T(t))_{t\ge 0}$ on the Hilbert space $X$, and $D\in\mathcal{L}(X)$. I want to show that the operator $A+D-2\|D\|I$ with domain ...
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1answer
23 views

Is this a metric on $\mathbf P\mathcal H$?

Let $\mathcal H$ be a real or complex Hilbert space with inner product $\langle\cdot,\cdot\rangle$. On the projective space $\mathbf P\mathcal H = \left(\mathcal H\setminus\{0\}\right)\big/{\sim}$ ...
2
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1answer
42 views

Powers of compact operators

Consider a Hilbert space $H$ and a compact self-adjoint operator $T : H \to H$. I want to prove that all positive powers (especially fractional powers) of $T$ are compact. From the spectral theorem, I ...
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18 views

an objective question from functional analysis [on hold]

Let $A$ and $B$ be bounded operators on a Hilbert space $H$ such that $AB=BA$. Let $\lambda$ be an eigenvalue for $A$. Then it must be that a)$B$ has no eigenvalue b)$B$ has at least one ...
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2answers
30 views

Prove Operator is a Projector

Let $\mathscr{H}$ be a complex Hilbert space. A projector is a linear map $P:\mathscr{H}\to\mathscr{H}$ such that $P\circ P = P$. I'm trying to prove the following claim, from the information given ...
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1answer
21 views

Z-transform and$ H_2$ space

The following is from the preliminaries of a paper. Let $\mathbb{D} = \{z \in \mathbb{C} : |z|<1\}$ be the unit disc of complex numbers. A function $G:(\mathbb{C} \cup \{\infty\})\backslash ...
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0answers
19 views

Hamiltonian: Scattering Spaces

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a family of projections: $$1(r)^2=1(r)=1(r)^*\quad(r\geq0)$$ Denote for shorthand: ...
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2answers
188 views

What's the connection between Banach/Hilbert spaces and tools like power series, Fourier transforms etc.

I've learned, abstractly, about Banach and Hilbert spaces, and more concretely about $l^p$ and $L^p$ spaces. I also understand these ideas have something to do with a variety of tools that are useful ...
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1answer
23 views

Proving an integral identity

I'm dealing with the Hermitian operator, and I've been asked to prove that all $f(x) = x^n e^{\alpha x}$ belong to $L^2(-\infty,\infty;e^{-x^2/2})$ by showing that: $$\int_{-\infty}^{\infty}x^m ...
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0answers
17 views

Lemma 3.3-7 and Theorem 3.6-2 in Kreyszig's “Introductory Functional Analysis With Applications”: What if completeness is lost? [duplicate]

Let $X$ be an inner product space, and let $M$ be a non-empty subset of $X$. Then we have the following: (a) If the space of $M$ is dense in $X$, then $M^\perp = \{0 \}$, that is, $x \in X$, $x ...
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1answer
27 views

On the subset of a closed vector subspace

Theorem: Let $H$ be a Hilbert space, and let $U$ and $V$ be closed subspaces of $H$ such that $U\subset V$. Then there exists a nonzero vector $v\in V\backslash U$ such that $v\bot U$. The fact that ...
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1answer
31 views

Need help understanding compact embedding of hilbert spaces

I am trying to understand the following statement, and I would like some clarification Consider a Hilbert space $H$ which is compactly embedded in a Hilbert space $L$, with $H^*$ being the dual ...
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1answer
18 views

Norm of a linear continuous form

Let $E=\{f\colon[0,2]\to\mathbb{R} \mid f \text{ continuous} \}$ be a prehilbert space equipped with inner product: $$\langle f,g\rangle=\int_0^2 f(t)g(t)\, dt$$ And let : $$U\colon E ...
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1answer
19 views

Theorem 3.3-1, Lemma 3.3-2, and Theorem 3.3-4 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: How to write these as one?

I'm trying to prepare some ancilliary material on the following three results in sec. 3.3 in the book Introductory Functional Analysis With Applications by Erwine Kreyszig: (First, I'm giving ...
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2answers
44 views

Intersection of Hilbert spaces

Consider two Hilbert spaces $H_1$ and $H_2$ with inner products $\langle \cdot,\cdot\rangle_1$ and $\langle \cdot,\cdot\rangle_2$ generating norms $\Vert \cdot \Vert_1$ and $\Vert \cdot \Vert_2$ ...
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1answer
35 views

Prob. 2, Sec. 3.3 in Erwin Kreyszig's “Introductory Functional Analysis With Applications”: How to minimise the norm?

Let $z$ be a given complex number. Let $M \subset \mathbb{C}^n$ be given by $$M \colon= \left\{ (\xi_1, \ldots, \xi_n ) \in \mathbb{C}^n \mid \sum_{i=1}^n \xi_i = z \right\}.$$ Then $M$ is convex ...
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1answer
62 views

Spectral Measures: Core Lemma

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a dense domain: ...
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1answer
18 views

Reducing Subspaces: Hamiltonian

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a projection: $$P\in\mathcal{B}(\mathcal{H}):\quad P^2=P=P^*$$ Then one has: ...
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29 views

Hamiltonian: Invariant Core

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ And an operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$ Denote its evolution by: ...
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14 views

Operator norm of symmetric Matrix in Hilbert Space with Hermitian Inner Product

Assume we have a postive definite real matrix $P$ and we define an inner product on a finite dimensional hilbert space $\langle x, y \rangle = x^\top P y$ and clearly the induced norm is $|| x || = ...
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22 views

Is there a pseudocontractive mapping that is not strictly pseudocontractive?

Given a Hilbert space $H$, a mapping $T:H\rightarrow H$ is said to be pseudocontractive if $$\|Tx-Ty\|^2\leq \|x-y\|^2+\|(x-Tx)-(y-Ty)\|^2\,\,\, \forall x,y\in H,$$ and it is strictly ...
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1answer
67 views

Hamiltonian: Derivative

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ and an operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$ Denote for shorthand: ...
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1answer
19 views

Selfadjoint Operators: Weak Convergence

Given a Hilbert space $\mathcal{H}$. Consider a selfadjoint operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$ Regard a sequence: ...
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1answer
39 views

The property of closed subspace

We know that a set is closed if and only if every convergent sequence with elements in the set has a limit point in the set. I am reading a paper, and the paper claims that the following is due to S ...
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2answers
35 views

Show $ \langle Tx,x \rangle \in \mathbb R$ for all $x \in H$ implies $T$ is self-adjoint

Show that a linear operator $T: H \rightarrow H$ is self adjoint if and only if $\langle Tx, x \rangle \in \mathbb R$ for all $x \in H$. You may use that the equality that for all $x,y \in H$ ...
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1answer
86 views

Unique trace on a type $II_1$ von Neumann Algebra

Let $M \subseteq B(H)$ be a type $II_1$ von Neumann Algebra. Then any two non-zero ultraweakly continious normalised traces $Tr,tr : \rightarrow \mathbb{C}$ are equal. I'm trying to understand this ...
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1answer
12 views

A rescaled inner product inequality

I was wondering if the following inequality is true: Let $\xi_1,...,\xi_n$ be vectors in a Hilbert space $H$ and let $x_{i,j}$ be complex numbers such that $\prod x_{i,j}$ is real and $$\prod ...
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1answer
23 views

A question on Isometry between the orthogonal subspaces of Hilbert spaces

I was reviewing my class-notes on Functional analysis and the professor had mentioned that given a closed proper subspace $U$ of an hilbert space $\mathcal{H}$, $\exists $ a closed subspace ...
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1answer
43 views

A nonzero compact operator on a Hilbert space has a nonzero eigenvalue

Let $T:H\to H$ be a compact operator on a Hilbert space $H$, with $T\neq0$. Prove that $\exists c\neq0$ and $x\neq0$ such that $Tx=cx$. I was trying to prove it using the fact that if $T$ is a ...
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1answer
80 views

$\sin$ and $\cos$ are the basis of what space?

When learning Fourier expansions, we learn that $\{\sin(mx), \cos(mx)\}_{m \in \Bbb N}$ is an orthonormal basis for our space and thus we can expand functions in it. My question is what space is this ...
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23 views

Cauchy sequence in reproducing kernel Hilbert space

Consider a positive definite kernel $K:\mathbb N\times \mathbb N\rightarrow \mathbb R$. Denote the unique RKHS associated with $K$ by $\mathcal H_K$. The RKHS $\mathcal H_K$ consists of \begin{align} ...
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36 views

$P$ and $Q$ are unitarily equivalent iff dimensions of ranges and kernels are the same

Two projections $P,Q$ are unitarily equivalent if and only if $$dim(randP)=dim(ranQ)$$ $$dim(kerP)=dim(kerQ)$$ How can we show this? One directionn seems easy: If $P$ and $Q$ are unitarily eqv, ...
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1answer
63 views

How do I prove a differential operator has no purely imaginary eigenvalues?

Anyone who has taken a course in linear algebra knows how to prove the eigenvalues of a self-adjoint operator are real or the eigenvalues of a skew-self-adjoint operator are purely imaginary. This is ...
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63 views

Prob. 8, Sec. 3.5 in Erwin Kreyszig's Introductory Functoinal Anlaysis With Applications

Erwin Kreyszig's Introductory Functoinal Anlaysis With Applications Prob. 8, Sec. 3.5 $\DeclareMathOperator{\span}{span}$Let $(e_k)$ be an orthonormal sequence in a Hilbert space $H$, and let $M = ...
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2answers
44 views

A question about Fourier coefficients.

Is it true that the sequences $ (A_{n})_{n \in \Bbb{N}} = (0)_{n \in \Bbb{N}} $ and $ (B_{n})_{n \in \Bbb{N}} = \left( \dfrac{1}{\sqrt{n}} \right)_{n \in \Bbb{N}} $ are the Fourier coefficients of ...
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0answers
9 views

Existence of Schauder base for given operator

Suppose $A: l_2 \rightarrow l_2$ is a finite-rank linear bounded operator of dimension $k$. Is it true that there exists a Schauder orthonormal base for which only first $k$ columns will be nonzero ...