Questions tagged [hilbert-spaces]

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

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21 views

Why the initial space of $U$ is $\mathcal M\ $?

$\textbf {Polar decomposition}$ $:$ Let $\mathcal H$ and $\mathcal K$ be Hilbert spaces. Then any operator $T \in B(\mathcal H, \mathcal K)$ admits a decomposition $T = UA,$ where $U \in B(\mathcal H, ...
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8 views

End, $\mathcal{B}$, $\mathcal{L}$. Notation for endomorphisms

I am writing some notes and would like to ask if dealing with algebra $*$-representations, Hilbert spaces, etc. all finite dimensional, I should use $\mathrm{End}$ or $\mathcal{B}$ or $\mathcal{L}$ ...
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20 views

Clarification: $\{x^{\alpha}\}$ total in $L^2[a,b]$.

I've come across the following remark without many details: For $\alpha \in \{0, 1, \dots\}$, the sequence $\{x^{\alpha}\}$ is total in $L^2[a,b]$. By total, I mean $X = \{x^{\alpha}\}$, then $\...
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1answer
17 views

$L^{P} ( \Omega ; W^{1,P}(Y))$ can someone define this space. $L^{P}$ and $W^{1,P}$ are sobolve and Lp spaces. How norm can define?

I have this space $L^{P} ( \Omega ; W^{1,P}_{per}(Y))$ can someone define it. As far I understands it takes a function f from $L^{P}(\Omega)$ and maps it to Sobolov space with periodicity $Y$. Or $f : ...
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1answer
23 views

Can we say that $T^n$ is a positive operator?

Let $\mathcal H$ be a complex Hilbert space and $T \in \mathcal L (\mathcal H)$ be a positive operator. Can we say that $T^n$ is also a positive operator? I am able to prove that if $n$ is even then $...
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1answer
27 views

In a Hilbert Space: Minimal $\Leftrightarrow$ Biorthogonal

Perhaps obvious, but I'm struggling a bit with (i) $\Rightarrow$ (ii) only. Proposition. Let $\{x_n\}_{n \in \mathbb{N}}$ be a sequence in a Hilbert space $H$. Then the following are equivalent: i) ...
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2answers
11 views

Continuously applying an isometry operator on to a Hilbert space

While looking at the proof of https://en.wikipedia.org/wiki/Wold%27s_decomposition#A_sequence_of_isometries, I encounted that if $V \in B(H)$ is an isometry on a Hilbert space $H$. Then we have $V^n H ...
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1answer
30 views

Understand rank-one operators in Hilbert space

Let $\{e_n \}$ be an orthonormal basis on a Hilbert space $H$. When we say that $T$ is a rank-one operator, it means that the range of $T$ has dimension $1$. Then for example, would $Te_n=e_n$ be of ...
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15 views

Comparing decompositions for two non-orthogonal bases in Hilbert Spaces

Let $H$ be a real separable Hilbert space and $B^1=\{w^1_j\}_{j \in J}$ and $B^2=\{w^2_j\}_{j \in J}$ two basis sets. Consider the linear maps \begin{align} P^i_j: H& \longrightarrow \mathbb{R}\\ ...
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39 views

Knowing the number of negative eigenvalues of a $ A $ operator, is it possible to know of the $ -A $?

Let $H=(H, (\cdot, \cdot)_H)$ be a Hilbert space and $A: D(A) \subset H \longrightarrow H$ be a linear, densely defined and self-adjoint operator. Suppose that the spectrum $\sigma(A)$ of $A$ is ...
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40 views

By Bessel's inequality, $\sum_{n=1}^\infty |c_n|^2 \leq \|f\|^2 < \infty$, Why $\|f\|^2 < \infty$

Suppose that $\varphi_n$ is an orthonormal sequence in a Hilbert space $H.$ Let $$V_N = \text{span}\{\varphi_1, \varphi_2, ..., \varphi_N\}, \ V = \cup_{N=1}^\infty V_N$$ ($V$ is the vector space of ...
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1answer
20 views

Showing compactness that involves unilateral shift

Let $V:l^2(\mathbb{N})\to l^2(\mathbb{N})$ be the unilateral shift, the unique bounded operator on $l^2(\mathbb{N})$ that satisfies $$ Ve_n=e_{n+1} $$ , where $(e_n)_{n\in \mathbb{N}}$ is an ...
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18 views

p-summable sequences

I am new to functional analysis. So please don't mind if my questions are little bit obvious/doesn't make sense. Suppose we have a p-summable sequence $(a_n)$. We know by definition that $$\sum_{n=1}^\...
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1answer
15 views

principal eigenvalue in infinite dimensions

For $A$ a symmetric, positive let $\lambda_1$ be its principal eigenvector. I want to show that $$\lambda_1=max_{v\neq 0}\frac{<Av,v>}{||v||^2}$$ I found this link: Prove that a $2\times2$ real ...
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12 views

Is the diagonal less than a hermitian matrix?

Suppose that $M$ is a positve hermitian square matrix acting on the space $\mathbb{C}^n$. Denote by $e_1,...,e_n\in \mathbb{C}^n$ be the canonical basis. Thus, the diagonal of $M$ determines another ...
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21 views

A unitary operator is a closed operator

Let $\mathcal{H}$ be an infinite-dimensional Hilbert space, with $U$ a densely defined, unitary operator. I was wondering if such an operator is in fact a closed operator, which is equivalent to \...
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5 views

Orthogonal projections and semigroup exponentially stable

I am studying semigroups with the property that are exponentially stables. I have the following exercise: If $T(t)$ is a semigroup analytic and exponentially stable on a Hilbert space $X$ and $\{ Pr_k ...
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1answer
21 views

Show that for the two sided left shift operator in $\ell_2(\mathbb{Z})$ has $\{|\lambda | =1\} $ in its spectrum

Let $S:\ell_2(\mathbb{Z}) \to \ell_2(\mathbb{Z})$ be the left shift operator $S(\ldots, x_{-2}, x_{-1}, x_0, x_1, x_2, \ldots) = (\ldots, x_{-1}, x_0, x_1, x_2, x_3 \ldots)$ I want to show that $\{|\...
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20 views

If $T \in L(X,X)$ is a bounded operator and $lim_{n \to \infty} ||T^n|| = 0$ then $\sigma(T) \subset B_r(0)$ for some $r < 1$

Let $X$ be a Hilbert space and let $T: X \to X$ be a bounded operator such that $lim_{n \to \infty} ||T^n|| = 0$. I need to show that there exists $r < 1$ such that $\sigma(T) \subset \overline{B_r(...
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19 views

Is the composition of closed unbounded operators closed?

Let $H$ be a complex Hilbert space with linear subspaces $U,V$. Then a (not necessarily bounded) linear operator $T:U\to V$ is said to be closed if $$\text{Graph}(T)\equiv\lbrace(u,Tu):u\in U\rbrace$$ ...
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21 views

Operator valued polynomials

I was reading Reed & Simon, where the following theorem is stated and proved. Theorem: Let $A_{k}$ be a self-adjoint operator on (a Hilbert space) $\mathscr{H}_{k}$. Let $P(x_{1},...,x_{N})$ be a ...
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38 views

Can a bounded and continuous function on a finite open interval be extended to its closure?

Let $f \in C(I; H)$, where $I = (a,b)$ is a finite open interval of $\mathbb{R}$ and $H$ is a Hilbert space. Also, $f$ is bounded on $I$. Then, can $ f$ be extended to $\bar{f}$ such that $\bar{f} \in ...
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48 views

Find the spectrum of self-adjoint operator $T$ that satisfies $T^3+T^2+T+I=0$

Let $H$ be a Hilbert space, and let $T:H \to H$ be a bounded operator such that $T$ is self-adjoint and also it satisfies the equation: $$T^3+T^2+T+I=0$$ I want to find the spectrum $\sigma(T)$ By ...
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25 views

Hilbert Space Operator

I am new to functional analysis and I stumbled upon a problem from Hilbert Spaces where an operator was defined as below: $$T_n(x) := (x,\phi_n)\phi_1$$ where $x \in H$ and $(\phi_n)_{n \in N}$ is the ...
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1answer
31 views

Show that $\{P_n\}_{n \geq 1}$ does not converge in norm topology.

Let $\mathcal H = \ell^2 (\Bbb N)$ and $P_n$ be the projection onto $\text {span}\{e_0,e_1, e_2, \cdots, e_n \}.$ Show that the sequence $\{P_n\}_{n \geq 1}$ does not converge in norm topology. ...
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1answer
19 views

Inner product of elements in an orthonomal set and elements in a Hilbert Space

Let $\{e_k : k\in K \}$ be an orthonormal set in a Hilbert space $H$, where $K$ is some arbitrary set. Let $x \in H$. Show that for each $n\in \mathbb{N}$, $F_n = \{ k\in K: |\langle e_k, x\rangle|^2 ...
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1answer
14 views

Show that every eigenvalue to a unitary operator has absolute value 1

Let $A$ be a unitary operator on hilbert space $H$, i.e. $$(Au|Av) = (u|v)$$ for all $u,v \in D_A.$. I'm asked to show that all eigenvalues to this unitary operator has absolute value 1. My attempt: ...
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24 views

Hilbert Space, anda Bessel’s inequality

could you please help me in this matter question Let $H$ be a Hilbert space and $B=\{e_{n}\}_{n \in \mathbb{N}} \subset H$ be a sequence of orthonormal vectors. Prove that $e_{n} \rightharpoonup 0$ I'...
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1answer
36 views

Elegant proof of pre-annihilator equal to orthogonal complement

Let $X,Y$ be Hilbert spaces. I just proved that, given $A: X\rightarrow Y$ a bounded operator, one has:$$\ker(A^*)_\perp=\ker(A^t)^\perp$$ Here $U_\perp$ means pre-annihilator, $V^\perp$ means ...
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1answer
38 views

Unbounded operators, Hilbert spaces and Borel sets

Let $H$ be a Hilbert space and $\langle , \rangle$ its inner product. Let $X : $ be a self-adjoint linear operator and let $F_X : H \to \mathbb{R}$ be defined as $F_X(a) = \langle a , Xa \rangle$. Let ...
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58 views

Polynomial in Hilbert space definition

I ve been looking around but haven't really found anything. How does one define polynomials in a Hilbert space $\mathcal{H}$. So a function $F:\mathcal{H}\rightarrow \mathcal{H}$ such that $F$ is a &...
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1answer
25 views

Defining bounded operator on orthonormal basis.

Let $\{e_i : i \in I\}$ be an orthonormal basis for a Hilbert space $H$. Suppose we are given a collection of vectors $\{\xi_i\}_{i \in I}$ with $\sum_{i \in I}\|\xi_i\| < \infty$. I want to prove ...
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1answer
43 views

Sine Fourier coefficients of derivative

Let $f$ be a $C^1$-function on $[0,\pi]$ with $f(0)=f(\pi)=0$. We know that the family $(\sin(nx))_{n\in\Bbb N}$ is a complete orthonormal system on $L^2(0,\pi)$ (maybe up to some rescaling). Let $(...
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39 views

Can we turn $\mathfrak{X}(M)$ into a Hilbert Space/ Inner product space

Given a Compact, Riemannian Manifold $(M,g)$, I'm wondering if we can induce an inner product space on $\mathfrak{X}(M)$, the set of smooth vector fields on $M$ that might have some interesting ...
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2answers
75 views

Let H be a Hilbert space and $f : H →\mathbb{R}$ be defined by $f (x) = 1/2||x||^2$. Show that $∂f(x) =\{x\}$ for all $x$ in $H$. [closed]

Definition: Let $H $be a Hilbert space and $f : H →(-\infty,\infty]$. The subdifferential $\partial f(x)$ of a $f$ is defined by $\partial f(x) =\{w\in H:\langle w,y-x \rangle~\leq f(y)-f(x), \forall ...
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16 views

Direct product of Hilbert spaces

I know of direct sum in the category of Hilbert spaces. Is there a concept of direct product too, wherein one can consider arbitrary tuples ? (looks like a no, as in order to talk about convergence ...
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1answer
53 views

If $\operatorname{Im}(x) \perp \operatorname{Im}(y)$, then $\|x+y\| = \max\{\|x\|, \|y\|\}$.

Let $x,y \in B(H)$ be bounded operators on the Hilbert space $H$ such that $xy=yx = 0$ and $\operatorname{Im}(x) \perp \operatorname{Im}(y)$. Moreover, assume that $\operatorname{Im}(y)$ has finite-...
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22 views

Uniqness of Fourier expansion in a non-separable Hilbert space

All my life, I have been operating inside separable Hilbert spaces. Everything is all dandy as any orthonormal basis is a sequence. Let $e_n$ be the orthonormal basis of the Hilbert space $H$. I have ...
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21 views

Computing a resolvent by a density argument

I have a differential operator $L$ on $L^2(\mathbb{R})$ and I want to compute its resolvent set by solving the equation $$(L-\lambda)v=f\;\;\;(1)$$ for arbitrary $f$ in $L^2(\mathbb{R})$. I saw an ...
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29 views

Computing the resolvent set $L^2([0,\infty])$ by solving a non-homogeneous ODE

I am computing the resolvent set of an operator on $L^2([0,\infty])$. One way to do that is to solve the corresponding of an inhomogeneous ODE with inhomogeneous term being an arbitrary $f\in L^2([0,\...
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1answer
49 views

a convex closed set of a Banach space contains no closest point

I need to prove that: For the Banach space $E=C([0,1])$ with $||.||_{\infty}$ and the closed convex set $A=\{f \in E : f(1)=0 \}$, can we find an element $f \in E$ such that $A$ contains no closest ...
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1answer
31 views

Does $\|Tx\|=\|T^*x\|$ for all $x$ imply normality for real Hilbert spaces?

The proof for complex Hilbert spaces crucially relies on the fact that $\langle Sx,x\rangle=\langle Tx,x\rangle$ for all x implies that S=T. This fact fails in the case of real Hilbert spaces as early ...
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26 views

Prove that $\|T\| = \max \{\|T_0\|, \|T_1\| \}.$

Let $\mathcal H_0$ and $\mathcal H_1$ be two Hilbert spaces. Then $\mathcal H_0 \oplus \mathcal H_1$ is also a Hilbert space endowed with the inner product $\langle x_0 \oplus x_1, y_0 \oplus y_1 \...
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1answer
25 views

Will $P$ be necessarily bounded?

Let $\mathcal H$ be a Hilbert space. Let $P : \mathcal H \longrightarrow \mathcal H$ be a self-adjoint idempotent linear operator. Can $P$ be necessarily bounded? I can't prove boundedness of $P$ ...
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1answer
55 views

State on *-algebra: $\phi(|f+g|^2)$ when $\phi(|f|^2),\phi(|g|^2)>0$ for monomials

Let $\mathcal{A}$ be the unital *-algebra generated by $N^2$ projections $u_{i,j}=u_{i,j}^2=u_{i,j}^*$, such that the rows/columns are partitions of unity $\sum_k u_{ik}=1_\mathcal{A}=\sum_k u_{kj}$, ...
2
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1answer
51 views

Factor a rank one operator on a Hilbert C*-module.

Let $A \subset \mathcal{L}(\mathcal{H}_0)$ be a concrete $C^*$-algebra and $X$ be a right Hilbert A-module. For each $x,y \in X$ we have rank one operators \begin{align} \theta_{x,y}: X & \to X \\ ...
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1answer
53 views

Pure states on noncommutative $\mathrm{C}*$-algebra can “be disturbed”

In a finite dimensional commutative $\mathrm{C}^*$-algebra $A\subset B(\mathbb{C}^N)$, if a vector state $\varphi_x(f)=\langle x,f(x)\rangle$ given by $x\in P(\mathbb{C}^N)$ is pure, then there is no ...
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1answer
35 views

How to prove that frame functions on Hilbert spaces are additive?

Let $\newcommand{\calH}{\mathcal{H}}\newcommand{\eff}{\operatorname{Eff}(\calH)}\calH$ be some separable Hilbert space, and denote with $\eff$ the set of effects on $\calH$, that is, the set of ...
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2answers
32 views

$AA^{*}-I$ is negative semi-definite

I need to show $(AA^{*}-I)$ is negative semi-definite where A is a linear operator. I can show $AA^{*}+I$ is positive definite since $\langle x, (AA^{*}+I)x \rangle=\|A^{*}x\|^2+\|x\|^2>0$ for $x \...
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1answer
43 views

If $(f_n)$ is an orthonormal basis and $\lambda_n\to\infty$, is $T(t)x:=\sum_ne^{-λ_nt}\langle x,f_n\rangle f_n$ uniformly continuous?

Let $H$ be a complex Hilbert space, $(f_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ and $(\lambda_n)_{n\in\mathbb N}\subseteq(0,\infty)$ be nondecreasing with $\lambda_n\xrightarrow{n\to\infty}\...

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