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

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1answer
79 views

Spectrum of the operator $A(f,g)=(g,\Delta f-f)$

Let $\Omega$ be an open set in $\mathbb{R^n}$. We consider the product Hilbert space $H=H^1_0(\Omega)\times L^2(\Omega)$ with the norm $$|(f,g)|^2=\int_\Omega (|\nabla f|^2+|f|^2+|g|^2 ) dx$$ We ...
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0answers
65 views

Any finite-dimensional subspace of a Hilbert space is closed: easier proof?

A noted theorem is that a finite-dimensional subspace of a Hilbert space must be topologically closed. I have seen some proofs of this theorem which are less simple than this, but what is wrong with ...
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26 views

Question about convergence of sum

Let $T\in B(H,E)$ where $H$ a seperable hilbertspace, $E$ a seperable Banach space. By parsevals identity $$\left\|T^*\right\|^2= \sup_{ \left\|x^*\right\|\leq 1}\left\|T^*x^*\right\|^2 = \sup_{ ...
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1answer
45 views

Bound on number of mutually orthonormal eigenfunctions

Let $E$ be the vector space of real valued continuous functions on an interval $[a,b]$. Let $K = K(x,y)$ be a continuous function of two variables, defined on the square $a \leq x \leq b$ and $a \leq ...
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0answers
88 views

Skew adjoint operator with uncountable spectrum

Let $H$ be a Hilbert space. I just want an example of a skew adjoint operator $(A^*=-A)$ with uncountable spectrum. I also want an example for unbounded differential operators. The only example I ...
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1answer
59 views

Is this subspace dense in $L^{2}(\Omega,\mu)$

Let $(\Omega,\mu)$ be a measure space, and let $X=L^{2}(\Omega,\mu)$ be the complex Hilbert space of square-integrable complex measurable functions on $\Omega$. (Each $f \in L^{2}$ is an equivalence ...
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1answer
88 views

Show that $\|e^{tA}\| \le e^{t\|\Re (A)\|}$

Let $X$ be a complex Hilbert space, and let $A$ be a bounded linear operator on $X$. Define the real part of $A$ to be $\Re(A)=\frac{1}{2}(A^{\star}+A)$, and define ...
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1answer
77 views

$L$-Lipschitz gradient of $f$ implies inverse strongly monotone of $\nabla f$

Let $f:\; \mathcal{H} \to R$ be a continuously differentiable convex function such that $$\|\nabla f(x) -\nabla f(y)\|\leq L\|x-y\|.$$ Prove that the mapping $\nabla f$ is $1/L$ inverse strongly ...
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1answer
41 views

Subspace of certain series in a Hilbert space is compact

Let $E$ be a Hilbert space and let $\{x_{n}\}$ be an orthonormal basis.  Let $\{c_{n}\}$ be a sequence of positive numbers such that $\sum c_{n}^{2}$ converges.  Let $C$ be the subset of $E$ ...
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156 views

Does the shift operator on $\ell^2(\mathbb{Z})$ have a logarithm?

Consider the Hilbert space $\ell^2(\mathbb{Z})$, i.e., the space of all sequences $\ldots,a_{-2},a_{-1},a_0,a_1,a_2,\ldots$ of complex numbers such that $\sum_n |a_n|^2 < \infty$ with the usual ...
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1answer
63 views

Find the norm of functional

Consider the functional from $l_2$. $$ x=(x_n)\mapsto \sum \frac{x_n+x_{n+1}}{2^n}. $$ What is the norm of the functional?
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1answer
82 views

Convergence of a sequence of unit vectors in a Hilbert space

Let $E$ be a vector space (over $\mathbb{R}$) with a positive definite hermitian form and let $\{x_{n}\}$ ($x_{n} \not= 0$) be a sequence converging to $x$ in the $L^{2}$ norm ...
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2answers
179 views

Basic Quantum Mechanics Concepts with Continuous Spectra

The following are a couple excerpts of the first chapter of Sakurai and Napolitano, Modern Quantum Mechanics, 2nd edition: Prior to these formulas, the text discusses the fundamental mathematics ...
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1answer
61 views

Sufficient condition for two operators being identical on Hilbert space

Considering two bounded linear operators $S,T$ in $\mathcal{B}(X)$, where $X$ is a complex Hilbert space. If $\def\norm#1#2{\langle {#1},{#2}\rangle} \norm{Sx}{x} = \norm{Tx}{x}$ for all $x\in X$, do ...
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1answer
149 views

Trivial Orthogonal Complement $\implies$ Denseness!

Disclaimer: Though I don't need it anymore this is interesting in its own! Is it true that if the orthogonal complement is trivial then the subset was dense: $$A^\bot=\{0\}\implies\overline{A}=X$$ ...
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2answers
134 views

What's the spectrum of this operator in $\ell^2$?

Suppose that $\ell^2 = \biggl\{(x_n)_n \in \mathbb{K}^{\mathbb{N}_0} \biggm| \sum_{n=1}^{\infty}|{x_n}^2| < +\infty \biggr\}$ is a Hilbert-space with the inproduct $\langle\cdot,\cdot\rangle_2: ...
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1answer
62 views

Is projection on a convex closed weakly-sequentially continuos?

I think to have proved the following: Given K a convex closed(maybe also limited is needed)subset(also curve not just subspaces) of an Hilbert space H, is well defined the projection operator $p_K:H ...
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1answer
103 views

Please check proof of convergence in $L^2(0,T;L^2)$ of a composition (uses Nemytskii operator)

Suppose we have a continuous function $g:\mathbb{R} \to \mathbb{R}$ that satisfies $|g(x)| \leq C|x|$. Let $u_{n} \to u$ in $L^2(0,T;L^2)$. I want to show that $g(u_{n'}) \to g(u)$ in $L^2(0,T;L^2)$. ...
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1answer
111 views

Strong convergence of an “averaging” operator

Let $X$ be an Hilbert space and $S:X \rightarrow X$ be a bounded linear operator with $||S||=1 $ Define $$T_n= \frac{1}{n} \sum_{r=0}^{n-1} S^r$$ I want to show it converges strongly to some ...
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1answer
21 views

I need help showing something is a linear continuous operator.

Define $T:C([0,1])\rightarrow C([0,1])$ by $T(f)(x)=f(0)+\int_0^xtf(t)dt$ I want to show that $T$ is a continuous linear operator.Showing the linear part is easy enough, but I am not quite sure how to ...
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1answer
73 views

Concluding that a linear operator on a Hilbert space is invertible

Setting: Let $H$ be a Hilbert space with two inner products, $\langle \cdot,\cdot\rangle$ and $[\cdot, \cdot]$, and $S:H\to H$ be a bounded linear operator such that for all $x,y\in H$, we have ...
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1answer
34 views

About what happens to eigenspace under functional calculus for Unbounded Operator

Let $T$ be an unbounded self adjoint positive operator on a Hilbert Space $\mathcal{H}$. Let $x \in \mathcal{H}$ be a vector such that $Tx = x$. Is it true that $T^{\frac{1}{2}} x = x$. For what $f$ ...
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1answer
109 views

Extending a compact operator to the entire Hilbert space

In a course I'm taking we defined compact operators as a linear mapping $H\rightarrow H$, where $H$ is a Hilbert space, that maps bounded sets to relative compact ones. The lecturer mentioned that the ...
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2answers
59 views

Proof: adjoint map of projection is a projection and …

Let $V$ be a pre hilbert space and $\pi \in \mathrm{End}(V)$. Show: the adjoint map $\pi^+$ of a projection (meaning: $\pi^2 = \pi$) is a projection itself. Show then: a projection $\pi$ is ...
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1answer
57 views

Isomorphism between Hilbert spaces

I want to show that the function $$ L^2(\Omega,\mathcal{O})\longrightarrow L^2(\widetilde{\Omega},\mathcal{O}) \colon f \longmapsto f|_{\widetilde{\Omega}}$$ is a isomorphism, where ...
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1answer
50 views

Determining the exact form of a projection in a Hilbert space

Let $$\Omega = \left\{f(x) \in \mathcal{L}^2[0,T]: \frac{1}{T}\int_0^Tf(x)dx = \mu,~ a \le f(x) \le b,~\forall x \in [0,T]\right\},$$ where $\mathcal{L}^2[0,T]$ is the set of Lebesgue ...
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2answers
135 views

Find the minimum distance that equal maximum inner product

If $x_0 \in$ $H$ (Hilbert Space) and $M$ is a closed linear subspace of $H$, prove that $$\min \{\|x - x_0\|: x \in M\} = \max \{\langle x_0, y\rangle : y \in M^\perp, \|y\| = 1\}.$$ I suppose $P$ ...
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1answer
110 views

Prove that this space is not Banach

Let $\Omega\subset\mathbb{R}^n$ be an open, bounded set with boundary $\partial\Omega$ of class $C^1$. $$\mathcal{A}:=\{u\in C^2(\bar\Omega):u=0\text{ on }\partial\Omega \}$$ endowed with the scalar ...
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1answer
103 views

Normal Compact Operator: not diagonalizable!

To proposition 5.17 in Weidmann's 'Lineare Operatoren in Hilberträumen' (german version) it is noted that the expansion of compact operators that are normal rather than self adjoint doesn't apply in ...
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1answer
48 views

Verification of conclusions regarding duality maps

I have two conclusions drawn from two results. I want to know how valid these two conclusions are. Firstly Consider the duality mapping(set-valued) $J:X \rightrightarrows X^{*}$ defined: $J(u) := \{ ...
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2answers
197 views

Symmetric Operator vs. Real Spectrum

For symmetric operators one has a characterization: $$A\text{ symmetric}:\quad A=A^*\iff\sigma(A)\subseteq\mathbb{R}$$ (I want to investigate to what extend symmetry is a necessary assumption.) ...
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1answer
60 views

When $M$ is closed $M^\perp$ is one-dimensional vector space

If $M=\{ x: Lx=0\}$, where $L$ is continuous linear functional on $H$ (Hilbert Space). Prove that $M^\perp$ is vector space of one-dimensional unless $M= H$. I know $M$ is closed so that $M^\perp$ is ...
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1answer
129 views

Operator: not closable!

Is there an operator between Banach spaces with the following properties: $$T:\mathcal{D}(T)\subseteq X\to Y:\text{ injective, dense range, continuously invertible, not closable!}$$ (Note that the ...
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1answer
55 views

Maximal set at Hilbert space

I need to prove the following: If B orthonormal set in a Hilbert space X, then B is maximal if and only if $ B^{\perp} =\{0\} $ I tried the following : $ B^{\perp} =\{0\} $ => B maximal $ ...
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1answer
42 views

$T$ has a finite rank $\iff$ $\exists N \in \mathbb{N}$ such that $\lambda_n=0$, $\forall n \geq N$

The question goes as follows: $T$ has a finite rank $\iff$ $\exists N \in \mathbb{N}$ such that $\lambda_n=0$, $\forall n \geq N$. Given is the data: $X$ is a Hilbert space with an orthonormal ...
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74 views

A question on functional analysis

Let $H_i$, where $i = 1,2$ be Hilbert spaces and $T_i : H_i \rightarrow H_i$ be closed operators, such that $T_i$ have positive spectrum. Let $\phi : H_1 \rightarrow H_2$ is an isometric isomorphism ...
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1answer
43 views

$Te_n$ converging to zero

I have the following question in my functional analysis book I dont understand: $X$ is an infinite dimensional Hilbert space with an orthonormal basis $(e_n)$. Show that if $T \in K(X)$, then $Te_n ...
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42 views

Spectral theory - how to prove this lemma?

in Anver Friedman, Foundations of Modern Analysis I found a lemma (6.7.3): If A is a self-adjoint operator and $\{E_\lambda\}$ is a spectral family such that $A=\int_m^{M+\varepsilon} \lambda ...
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4answers
790 views

Square root of a Hermitian operator exists

There are a lot of questions here about square root operators, but none of them addresses the basic question of existence, and I didn't find a very beefy section in Wikipedia talking about this, so ...
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1answer
44 views

Proving $||f+g||\cdot||f-g|| \le ||f||^2+||g||^2$ in a Hilbert Space

Let $f$ and $g$ be vectors in a Hilbert space $H$. Show that $$||f+g||\cdot||f-g|| \le ||f||^2+||g||^2$$ My question is, do i have to rewrite $||f+g||$ as $\sqrt{\langle f+g,f+g\rangle}$ and same ...
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1answer
224 views

What is the dual space in the strong operator topology?

Let $X$ be a Banach space, the strong operator topology on the space of bounded linear operators $\mathcal{B}(X)$ is defined by the family of continuous semi-norms $A\to\|Ax\|$, $x\in X$. What is the ...
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215 views

Is there a concept of a “free Hilbert space on a set”?

I am looking for a "good" definition of a Hilbert space with a distinct orthonormal basis (in the Hilbert space sense) such that each basis element corresponds to an element of a given set $X$. Before ...
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1answer
26 views

$\displaystyle ||T||= \sup_{||f||=1} |\langle Tf,f \rangle|$ for complex hilbert spaces

Let $H$ be a hilbert space. Let $T:H\to H$ be a linear bounded operator prove that $\displaystyle ||T||= \sup_{||f||=1} |\langle Tf,f \rangle|$ Obviously $\sup_{||f||=1} |\langle Tf,f| \le ||T||$ ...
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55 views

Let $T$ be a bounded operator such that $<Tf,f>=0$ then $T=0?$

Let $H$ be a hilbert space. Let $T:H\to H$ be a linear bounded operator such that $<Tf,f>=0$ for all $f\in H$. It is necesarily true that $Tf=0 ?$ When I mean Hilbert space over a field ...
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1answer
54 views

Inner product on Hilbert Spaces

It's an open question. How could you define an inner product for a product of noncontable Hilbert spaces?
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162 views

Operators such that $\langle Ax,x \rangle=-\langle x,Ax \rangle$

Let $X$ be a Banach space. We consider the differential equation: $$x'(t)=Ax(t), \ \ \ t\in\mathbb{R}$$ where $A$ is a bounded operator on $X$. If $X$ is a Hilbert space, and $x(t)$ is a solution of ...
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1answer
99 views

Creation and Annihilation Operators: Norm Estimate

Given the Fock space: $$\mathcal{F}(\mathcal{h}):=\bigoplus_0^\infty\mathcal{h}^{n}\text{ with } \mathcal{h}^{n}:=\bigotimes_1^n \mathcal{h},\mathcal{h}^0:=\mathbb{C}$$ Define the creation and ...
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1answer
70 views

Positive Operator: Norm Estimate

In class we encountered the statement: $$H\geq C1\quad(C>0)\implies\|\mathrm{e}^{-\beta H}\|<1\quad(\beta>0)$$ How does one prove this? Moreover, what about the weakened version: $$H\geq ...
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3answers
242 views

Is this space a Hilbert Space?

I have a space of continuously differentiable functions on [a, b] with the dot product defined in this way: $ x \cdot y = \int_a^b \! [x(t)y(t) + x'(t)y'(t)] \, \mathrm{d}t. $ Is this space a Hilbert ...
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2answers
76 views

Positive operators in Hilbert spaces

Let $H$ be a Hilbert space. I am just asking if there's some reference which studies operators $A$ with this property: $$\left\langle Ax,x\right\rangle \geq0,$$ for all $x\in H$. And $Ax=0$ whenever ...