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

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Existence Adjoint Operator: Boundedness?

Context This would make the check on the GNS construction much more simple. Problem Given a Hilbert space $\mathcal{H}$. Consider a merely linear operator $A:\mathcal{H}\to\mathcal{H}$. Suppose ...
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1answer
45 views

perturbation by orthogonal projection

Let $G$ be an operator with discrete spectrum on Hilbert space $H$ such that $\ker G$ is different from $\{0\}$. Let $P$ be the orthogonal projection onto $\ker G$, and let $G_{0} = G+P$. My ...
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1answer
18 views

Finite sum $\sum_{r,k} p_kP_r(x_k)f(x_k)P_r(x_m)=f(x_m)$

Let $x_0,\ldots,x_n\in\mathbb{R}$ be $n+1$ arbitrary real points and $p_0,...,p_n>0$ be positive real numbers. Let $P_0,P_1,\ldots,P_n$ be polinomials such that $$\sum_{k=0}^n ...
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1answer
35 views

Partial Isometries: Characterizations

Any partial isometry satisfies: $$\Omega\Omega^*\Omega=\Omega$$ From this, one derives projections: $$\Omega^*\Omega,\Omega\Omega^*$$ Conversely, given projections: $$\Omega^*\Omega,\Omega\Omega^*$$ ...
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0answers
28 views

Scalar product of $L_2$ with $\mu(E):=\int_E gdx$

I read in Kolmogorov and Fomin's Элементы теории функций и функционального анализа (p. 400 here) that, if we define measure $\mu$ for $E\subset[-1,1]$ by $$\mu(E):=\int_E g(x)dx$$ where the integral ...
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1answer
19 views

Hilber transform on [0,1)

Let $\mathbb{T}=[0,1)$ and $H$ be a Hilbert transform on $L^p(\mathbb{T})$ when $2\leq p< \infty$. If $f$ is $L^p$ and $f_n$ is trignometric polynomial such that $f_n\rightarrow f$ in $L^p$ sense. ...
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1answer
40 views

Eigenvalues of an integral operator on $L^2[-1, 1]$

Find the eigenvalues of the integral operator $K: L^2[-1, 1] \to L^2[-1, 1]$ defined by $(Kx)(t) = \int_{-1}^1 (1 - 3t \tau)x(\tau) d\tau$. I began with the fact that eigenvalues must be values ...
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1answer
39 views

Orthogonal Complements of polynomials in $L^2[0,1]$

I have two very difficult questions in my home work in function analysis, in which I have two calculate the complements of the following sets, in $L^2[0,1]$: All polynomials in the variable $x^2$ ...
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19 views

$f$ square-summable on $X'\times X''$, $\varphi_m$ square-summable on $X'$ and $\int f\cdot\bar{\varphi}d\mu'$ square-summable on $X''$

Let $X:=X'\times X''$ be the product of measure spaces $(X',\mu')$ and $(X',\mu'')$, endowed with the Lebesge extension $\mu:=\mu'\otimes\mu''$ of product measure $\mu'\times \mu''$ defined by ...
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1answer
54 views

Bound the norm of the partial trace of an operator on a Hilbert space

Let $H=H_1 \otimes H_2$ a composite Hilbert space and let $A, B$ bounded linear operators on $H$, and we can assume they are trace class. Let $A_2$ we denote the operator on $H_2$ obtained by taking ...
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1answer
21 views

Convolution Operator and Integration Operator

I have some questions about the following two operators. A convolution operator $T$. If $k \in \mathcal L^1(\mathbb R)$, then $$f(x) \mapsto \int_{-\infty}^\infty k(x-y)f(y) dy: \mathcal L^2(\mathbb ...
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1answer
51 views

Closed Subspaces of Hilbert Spaces

I read the following statements. But I do not know how to show it or any example to support it. Could anyone provide some explanation and examples, please? Thank you! The subspace $C^\infty$ ...
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25 views

Basis for Finite Dimensional Hilbert Spaces

Verify that a Hilbert space orthonormal basis in a finite dimensional Hilbert space is the same as an orthonormal basis in the sense of linear algebra. Here is what I know. Hilbert space ...
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1answer
39 views

Bounded Linear functional is the orthogonal projection onto its range.

Suppose we have $P:H\to H$, where $H$ is a hilbert space and $P$ is bounded and linear. Assume that it satisfies $P^2=P$ and $P^*=P$ where $P^*$ is the adjoint. Show that $||P||\leq 1$, that ...
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1answer
45 views

Møller Operators: Unitary Equivalence

Reference This is taken out of M. Reed and B. Simon, Scattering Theory. Problem Given a Hilbert space $\mathcal{H}$. Consider a free Hamiltonian $H_0$ and a perturbed one $H$. Introduce the ...
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17 views

totality of subset of an inner product space

How can I show that, given a subset $M$ of an inner product space $X$: If $M$ is a total set, then, $M^\perp =\{0\}$? ($M^\perp $ is the orthogonal complement of $M$)
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Møller Operators: Absolutely Continuous Subspaces [closed]

Given a Hilbert space $\mathcal{H}$. Consider a free Hamiltonian $H_0$ and a perturbed one $H$. Introduce the Møller operators: $$\Omega^\pm(H,H_0):=\mathrm{s-lim}_{\tau\to\pm\infty}e^{i\tau ...
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1answer
24 views

Prove there cannot be an inner product which turns $l^p$ into an inner product space?

For all $1\leq p < \infty, \mbox{ }p$ is not equal to 2, prove there cannot exist an inner product that turns $(X,\|\cdot \|_p)$ into an inner product space; that is, prove that there cannot be ...
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1answer
22 views

Nonnormal Operator: Empty Spectrum

Are there operators on Hilbert space having empty spectrum? (Surely, for Banach spaces they do exists.) Necessarily, they must be closed and can't be normal.
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1answer
36 views

about weak convergence in $L^2(0,T;H)$.

Exercise Suppose $H$ is Hilbert space and $u_k$ converges weakly to $u$ in $L^2(0,T;H)$. Suppose further we have the uniform bounds $\mathrm{esssup}_{0≤t≤T} ||u_k(t)||≤C$. Then ...
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1answer
33 views

How does one diagonalise an operator that has exponential elements?

I asked this question before on the Physics StackExchange, but as one commenter noted I might have more luck here. So the question is: What is the diagonal form of the (density) operator $\hat\rho$, ...
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1answer
54 views

A Question on Compact Operators on Hilbert Space

I read this question which I have no idea how to start. Could anyone provide me with some detailed answer, please? Thanks. Suppose that a linear operator $F$ from a Hilbert space $\mathcal H$ to ...
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1answer
23 views

Hilbert transform on $L^p(\mathbb{T})$

Let $\infty >p\geq 2$, then for $f\in L^p(\mathbb{T})$ (here $\mathbb{T}=[0,1)$), show that for any real-valued trigonometric polynomial $f$, we have $H(f^2-(Hf)^2)=2fHf$. The hint is to use the ...
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1answer
34 views

Inner products of weakly convergent sequences

I have a weakly convergent sequence in $L^2(U)$ (for $U$ some bounded open domain with smooth boundary), $u_k\rightharpoonup u$. I want to show that there is a sequence $v_k\rightharpoonup v$, such ...
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2answers
40 views

Spectral Measures: Spectrum vs. Numerical Range

Given a Hilbert space $\mathcal{H}$. Consider a normal operator $N:\mathcal{D}(N)\to\mathcal{H}$. The goal here is to prove: $$\langle\sigma(N)\rangle=\mathcal{W}(N)$$ By a previous result one has: ...
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1answer
55 views

An inequality about operator norm

Let $H$ be a Hilbert space and $T\in B(H)$, with $T_{i}\rightarrow T$ in strong operator topology. Then can we prove that $\liminf_{i\rightarrow \infty}||T_{i}||\geq ||T||$ ?
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1answer
39 views

A sum of a closed subspace and a closed one-dimensional space is closed

I'm losing my mind over this question. For $H$ a Hilbert space, $A,B$ closed subspaces, and $B$ is of dimension $1$, I want to prove that $A+B$ is also closed. I'm looking for a straightforward ...
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1answer
21 views

How can I calculate the projection of a Hilbert space into a closed subspace?

I was woundering if there is an easy way to calculate the projection of a Hilbert space into a closed subspace. Obviously one could write $P:H->C$ that is given by $P(x)=d$ s.a $d=inf||x-v||$ for ...
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182 views

Can the composite of two projections really fail to be a projection?

Let $H$ denote a Hilbert space. For any closed subspace $C \subseteq H$, write $P_C$ for the orthogonal projection onto $C$. Then according to wikipedia, the composite $P_U \circ P_V$ needn't be a ...
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1answer
39 views

$L_2$ as a Hilbert space and $\ell_2$

I know that, if measure $\mu$, with which measure space $X$ is endowed, has a countable base, i.e. if for any measurable $M\subset X$ there exists a measurable set $A_k\in\mathscr{A}$, where ...
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0answers
42 views

Does the orthogonal projection theorem guarantees uniqueness of the projected space?

Given a Hilbert space $H$, and linear map $P:H \to H$ such that $P^2=P$ and for every $x\in H$ : $\|Px\| \le \|x\|$, there is a closed linear-subspace $M$ such that $P=P_M$, the projection on $M$. My ...
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1answer
43 views

Example of a subspace S of a Hilbert space such that S^(⊥⊥) does not equal S?

I try to find an example of a subspace S of a Hilbert space H such that S^(⊥⊥) does not equal S. I know that subspace cannot be closed as for closed subspaces S^(⊥⊥)=S holds true. Does there exist ...
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1answer
60 views

Proving that if $<Ax,x>=0$ for every $x$, then $A$ is the zero operator

I feel kind of dumb but I needed this little claim as a part of a proof I'm writing, and I figured out that I'd better just ask, since I could not find the correct algebraic manipulation needed in ...
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37 views

If $A$ is a compact diagonal operator, with diagonal $\{\alpha_n\}$, then $\lim_{n\to\infty}\alpha_n=0$.

Here is my question: If $A\in \mathscr{B}(\mathscr{H})$ is a diagonal operator with diagonal $\{\alpha_n\}$, show that if $A$ is compact, then $\lim_{n\to\infty}\alpha_n=0$. Here is what I have: I ...
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1answer
36 views

Prove $Tx=(r_1x_1, r_2x_2, r_3x_3,…)$ is compact, $T:l^2\to l^2$, $r\in l^2$

Here is my question: Fix $r=(r_1,r_2,...)\in l^2$. Define $T:l^2\to l^2$ by $$Tx=(r_1x_1, r_2x_2, r_3x_3,...)$$ Prove that $T$ is compact. Here is what I have, input would be appreciated: Let ...
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1answer
38 views

Dense convex proper subset of the Hilbert space $l_2$: $\{x|\sum x_i=0\}$ [duplicate]

Let's consider the space $l_2$ (all sequences $x$ with $\sum x_i^2 < +\infty$) and its subset $Z = \{x|\sum x_i = 0\}$. I want to prove that the closure of $Z$ is $l_2$, but I can't. I tried to ...
3
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1answer
34 views

Definition of unitary operators

Let $\phi, \psi \in \mathcal{H}$ be some element from a hilbert space $\mathcal{H}$ and $U$ a linear operator $U: \mathcal{H} \rightarrow \mathcal{H}$. Does $$ \forall \phi: \| U \phi \|^2 = \| \phi ...
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1answer
25 views

Spectral Measures: Commuting Operators

The questions are given below!! Theorem Given a measure space $\Omega$ and a Hilbert space $\mathcal{H}$. Consider a spectral measure $E:\mathcal{B}(\Omega)\to\mathcal{B}(\mathcal{H})$. Denote ...
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If $T = A + iB$, where $A$ and $B$ are self-adjoint operators on a Hilbert space $H$, then this is said to be a Cartesian decomposition of T

If $T = A + iB$, where $A$ and $B$ are self-adjoint operators on a Hilbert space $H$, then this is said to be a Cartesian decomposition of $T$ Compute $T^∗$ in terms of $A$ and $B$.
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WOT convergence to SOT convergence

Let $H$ be a Hilbertspace and $T_n \in B(H)$ a sequence of operators with $T_{n+1} \geq T_{n}$. I want to to show that if there is a self-adjoint $T\in B(H)$ with $T_n \stackrel{WOT}{\rightarrow}T$ ...
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42 views

Hilbert spaces of holomorphic functions

Could you please give me some examples of Hilbert spaces of holomorphic functions? Or even books or notes on Hilbert spaces of holomorphic functions? I need just a good number of examples and perhaps ...
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1answer
75 views

Hilbert cube is compact

Let $\{u_n\}_{n\in \mathbb N}$ be an orthonormal set in $H$ (Hilbert space). How prove that the set $\displaystyle Q=\{x\in H :\ x=\sum_{i=1}^{\infty}{c_nu_n}, \ \mbox{where} |c_n|\leq\frac{1}{n} \}$ ...
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117 views

Stone's Theorem Integral: Basic Integral

Disclaimer This thread has been renewed: Stone's Theorem Integral: Advanced Integral Problem Given a finite Borel measure $\mu$ and a Hilbert space $\mathcal{H}$. Consider a strongly continuous ...
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1answer
15 views

For continuous Linear functional show $L \notin (C[0,1] , || . ||_2)^* $

For $L: C[0,1] \to \mathbb{C}$ denote the linear functional by $L(f) = f(0)$. Show $L \notin (C[0,1] , || . ||_2)* $ If I were to show $L \in (C[0,1] , || . ||_2)^* $ I would need to show that $L$ is ...
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38 views

Uniqueness of $n$-th root in Hilbert space

Let $H%$ be a Hilbert space and $A \in \mathcal L(H)$, $A = A^*$, $A \geq 0$. Let $B = \sqrt[n]A$, where $n \geq 3$, i.e. $B \in \mathcal L(H)$, $B \geq 0$, $B^n = A$. How to show that such operator ...
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If $V\subset H\subset V^*$ is a Gelfand triple, which is the natural inner product on $V^*$?

is there any natural way to define a inner product on $V^*$? First we could consider Riesz isomorphism $\mathfrak{R}:V\rightarrow V^*$, and define $\langle F, G\rangle_{V^*}:=\langle ...
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1answer
23 views

Sequence of unitary l.i. vectors such tha the sequence converges weakly to a non-zero vector, but not strongly

Let $\mathcal H$ be an infinite dimensional Hilbert space and let $\{x_{n}\}_{n=1}^{\infty}$ be a sequence of unitary linearly independent vectors. I know, using Bessel's inequality, that if the ...
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Matrices with Continuous Indices

The components of a matrix $A$ can be written as $a_{ij}$. In Quantum we're starting to talk about a generalization where the indices are not elements of $\Bbb N$, but are instead continuous. Our ...
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2answers
25 views

Square root of a bounded operator in Hilbert space

Consider the power series expansion $$ \sqrt{1-z} = 1+\sum\limits_{k=1}^\infty c_k z^k, $$ converging absolutely in the ball $|z| \leq 1$. Let $H$ be a Hilbert space and $A \in \mathcal L(H)$ a ...
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2answers
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Prove that this integral operator is compact

Let $X,Y=L^2(0,1)$, $k\in C^0([0,1]^2)$. Define $$ K:X\to Y,\,\,\,\,\,Kf(x):=\int_0^1k(x,y)f(y)dy\,\,\,\,\forall\, f\in L^2(0,1). $$ I have to show that $K$ is compact. My idea is to prove that $K$ ...