# Tagged Questions

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Can a selfadjoint operator have empty spectrum? (As far as I remember, yes; but just to be sure.) The point is that if so then the closure of its spectrum cannot equal the convex hull of its ...
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### Continuous linear image of closed, bounded, and convex set of a Hilbert Space is compact

Is my proof of this proposition correct ? And is this proposition well known? Proposition: Let $C$ be a closed, bounded, and convex set in a separable Hilbert space $H$. Let $L : H \to \mathbb{R}^n$ ...
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### Is the distance between disjoint closed convex subsets of a Hilbert space positive? Is it attained?

Let $H$ be an infinite dimensional and separable Hilbert space. Let $A,B$ be infinite, closed and convex subsets of $H$. If $A$ and $B$ are disjoint and if at least one of them is bounded, is the ...
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### Commutant of a set of operators and norm topology.

In the references I have it's remarked that the commutant $S'$ of a set $S$ in $B(H)$, where $H$ is a Hilbert space, is closed in the weak operator topology. And this is true because if ...
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### Hilbert subspaces of $B(\mathbb{R}^n)$

Apart from the one-dimensional subspaces, what are the Hilbert subspaces of $B(\mathbb{R}^n)$? I'm not even sure if such subspaces exist.
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### Category of Hilbert Spaces

Is it possible to triangulate the category of Hilbert spaces and bounded linear operators? I assume that one candidate for triangulation is the double dual space. What is a fact is that this ...
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### Kernel of the Extension of a Bounded Linear Operator

Suppose $T\colon E\to F$ is a bounded linear operator between Banach spaces. Moreover let $i\colon E\to E’$ be a dense, compact inclusion of $E$ into some other Banach space $E’$. Finally assume that ...
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### Chain of closed subspaces in a Hilbert Space

Let $H$ be a separable Hilbert Space (WLOG, we may assume $H=\ell_2(\mathbb{N})$ is the space of square summable sequences). Can there exist an uncountable chain of closed subspaces? In other words, ...
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Let $H$ be a real Hilbert space, $T:H\to H$ be a compact operator. Suppose that for every $x\in H$, sequence $(T^n x)_{n\in \mathbb{N}}$ converges weakly to $0$. How to prove that $... 0answers 70 views ### Conditions for the commutator of two operators on a Hilbert space to not be a nonzero scalar operator I have shown a proposition: Suppose$A$and$B$are two linear operators on a (complex) Hilbert space, where the domains may not be the whole. Then, if either$A$or$B$is normal and has an ... 1answer 33 views ### Show that the trace class operators on a Hilbert space form an ideal Let$(H, (\cdot, \cdot))$be a separable Hilbert space over$\mathbb{L} = \mathbb{R}$or$\mathbb{C}$. Suppose that$\{\phi_n\}_{n=1}^\infty$is an orthonormal basis for$H$. Let$\mathcal{B}(H)$... 1answer 44 views ### For a Hilbert space$\mathcal{H}$, is every bounded linear operator on$\mathcal{H}$a linear combination of unitary operators? Let$(\mathcal{H}, (\cdot, \cdot))$be a Hilbert space, and let$B \in \mathcal{B}(H)$be a bounded linear operator on$H$. If$\mathcal{H}$is a complex Hilbert space, then$B$can be written as a ... 1answer 36 views ### Prove that a Hilbert space is convex of power type$2$Let$X$be a Banach space. For$\epsilon \in (0,2]$, define: $$\delta_X(\epsilon) = \inf_{x,y \in X}\{1 - \|\frac{1}{2}(x + y)\| : \|x\| = \|y\| = 1, \|x-y\| \ge \epsilon\}.$$ Then we say that$X$... 1answer 28 views ### Prove that a linear and continuous operator admits inverse in Hilbert space Let$(H,(\cdot,\cdot))$an Hilbert space and$A:H\rightarrow H$a linear and continuous operator such that there exists$\alpha >0$such that $$(Au,u)\geq \alpha \|u\|^2 \text{ for each } u\in H.$$ ... 1answer 25 views ### Skew-adjoint differential operator$B$with spectrum$\sigma(B)=i(-\infty,-1]$Consider the Hilbert space$X=L^{2}\left(\mathbb{R}^n\right)$and the Schrödinger operator$A=i\Delta$defined on the domain$D(A)=H^2(\mathbb{R}^n)$. It is known that the spectrum of$A$is ... 0answers 49 views ### Inseparable Hilbert space and uncountable orthonormal basis construction I need help with exercise 13 from Methods of Modern Mathematical Physics I by Simon and Reed, chapter 2. Using direct sums, construct an inseparable Hilbert space and an uncountable orthonormal ... 1answer 32 views ### Hilbert Space: Weak Convergence implies Strong Convergence This probably might be a duplicate - let me know if so. I read the following in Graf's notes on quantum mechanics - can you give me a hint for the proof. In Hilbert spaces weak convergence in a way ... 0answers 43 views ### Prove that operator is completely continuous Let's consider Banach space$\ell^\infty$of bounded sequences$x = \{ \xi_n\}_{n=1}^\infty$: $$||x|| = \sup_{n \in \mathbb N} |\xi_n|.$$ Suppose matrix$||a_{i j}||_1^\infty$specifies operator$A$... 1answer 21 views ### Riesz (Hilbert-space) representation theorem and dirac delta on$\mathcal{C}_{0}$I am thinking about this for a while now, but don't get near an understanding, so I must have gotten something important wrong. I look at$\mathcal{C}_{0}$, the space of countinuous (bounded) ... 1answer 37 views ### Relation between$\epsilon$-pseudospectrum of operators If$H$is a Hilbert space and$\sigma_{\epsilon}(T)$denotes the space of all$\epsilon$-pseudospectrum of the operator$T$and$S, T\in B(H)$be such that$TS=ST=0$, why ... 0answers 26 views ### difference between uniformly convex norms and strictly subadditive norms? What is the difference between uniformly convex norms and strictly subadditive norms? why we need to define two above concept? how they help us to study Banach spaces? Is the norm induced by ... 1answer 64 views ### Formulas for Schrödinger unitary groups of operators Let$\Omega$an open set of$\mathbb{R}^n$. Consider the Hilbert space$X=L^{2}\left(\Omega\right)$and the Schrödinger operator$A=i\Delta$defined on the domain$D(A)=H^2(\Omega)$. Is there any ... 3answers 59 views ### Prove that if$T=T^*$and$\sigma(T)=\{\lambda\}$, then$T=\lambda I$Show that if$T$is a self adjoint linear operator on a Hilbert space such that the spectrum contains a single point$\lambda$, then$T=\lambda I$. Then, show this is false if$T$is not self ... 1answer 41 views ### Global bounded solution of$u_{tt}=\Delta u-mu+h$in the Hilbert space$X=H_{0}^{1}\left(\Omega\right)\times L^{2}\left(\Omega\right)$Let$\Omega$be an open subset of$\mathbb{R^n}$. Consider the linear wave equation $$\begin{cases} \dfrac{\partial^{2}}{\partial t^{2}}u\left(t,x\right)=\Delta ... 1answer 51 views ### Maximal subspace on which an operator is bounded Consider the Banach space X=C[0,1] of real continuous function on [0,1] equipped with the supremum norm. Consider the operator A:D(A)\to X, Af=f' for each f\in D(A)=C^1[0,1]. We can see that ... 2answers 60 views ### Properties of reflexive Banach spaces I just want to see the importance of reflexive Banach spaces and what is special about them compared to other Banach spaces. What kind of properties hold in reflexive spaces that do not necessarily ... 1answer 68 views ### Show that a subspace of l2 is not complete I would like to know if this exercise is correct. Let \Bbb R^\infty=\{x:\Bbb N\rightarrow \Bbb R: \exists n \text{ such that}\quad x(k)=0 \quad \forall k\geq n\}. Show that (\Bbb R^\infty, \| ... 2answers 67 views ### Properties of a set in \ell^2 space Let \ell^2 = \{x= (x_1,x_2,x_3,\ldots): x_n\in \mathbb C\text{ and } \sum_{n=1}^\infty |x_n|^2 < \infty\} and e_n \in \ell^2 be the sequence whose n-th element is 1 and all other elements ... 2answers 46 views ### Strongly continuous semigroup of operators which cannot be extended to a group Let X be a Banach space. We call a family of bounded operators (T(t))_{t\in \mathbb{R}} a strongly continuous group if it satisfies the properties of the strongly continuous semigroup but for ... 0answers 37 views ### Prove |(f, g)| \leq \int |f \bar g| for Complex Cases Let f, g be \mathbb C-valued functions defined on \mathbb R and f, g \in L^2. To prove the inequality in this title, I proceed as follows but got a weaker bound. Recall that \mathrm{Re}\ a ... 2answers 85 views ### Physical interpretation of L1 Norm and L2 Norm In signal analysis, students have no qualms about associating the L2 norm of a square integrable function f(t) as the energy associated with that signal. A good understanding of whether a function ... 0answers 23 views ### Helffer-Sjöstrand-Formula: Idea behind? I have to present the Helffer-Sjöstrand-Formula. Now I'm wondering: Why does it include a factor \chi(y\langle x\rangle^{-1}) for some bump function \chi and the chinese symbol ... 1answer 30 views ### Is there exists linear algebra basis for L^2[0,1] such that every element of it has length one and every two different element of it is orthogonal? We know by using axiom of choice every vector space over a division ring ( consequently any field ) has a basis like \mathbb E in the meaning of linear algebra ( \mathbb E is linear independent ... 0answers 39 views ### Proving that a Sturm-Liouville problem is in the limit-point/-circle case I would like to understand techniques anybody is able to detail to me on how one may actually prove that a particular Sturm-Liouville (S-L) problem, i.e., of the form ... 1answer 103 views ### Reproducing Kernel Hilbert Space (RKHS) constructed by the summation of positive-definite kernels. Let k_1,\ldots,k_p be positive definite kernels defined on \mathcal{X}\times\mathcal{X}, where \mathcal{X} is a non-empty set. k_i is the reproducing kernel of the Reproducing Kernel Hilbert ... 1answer 21 views ### Orthonormal basis L^2(a,a+2\pi) Let$$\mathcal{B}=\left \{\frac{1}{\sqrt{2\pi}},\frac{\cos x}{\sqrt{\pi}},\frac{\sin x}{\sqrt{\pi}},\frac{\cos 2x}{\sqrt{\pi}},\frac{\sin 2x}{\sqrt{\pi}},\dots\right \}$$. This is an orthonormal basis ... 1answer 60 views ### Sequence of orthogonal vectors in a Hilbert space Let$\{x_n\}_{n\in\mathbb{N}}$be a sequance of pairwise orthogonal vectors in a Hilbert space$H$. Show that the following are equavalent: (a)$\sum_{n=0}^\infty x_n$converges in the norm topology ... 1answer 32 views ### Weak convergence and infinite sum Suppose that$\psi_n$converges weakly to$\psi$in a Hilbert space$H$. Assume further$\{\phi_k\}$is an orthonormal sequence in$H$. Is it plausible that ... 1answer 57 views ### Is a linear operator on$\ell^2$defined by the inner product necessarily bounded? [duplicate] If$a=\{a_n\}\in \ell^\infty(\mathbb{R})$and$\langle a,x \rangle<\infty$for all$x\in \ell^2(\mathbb{R})$, (where$\langle a, x\rangle=\displaystyle \sum_{k=1}^\infty a_kx_k$), then is$a\in ...
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I'm stuck on an exercise about the spectrum of a selfadjoint operator on a Hilbert space. The problem is the following: Let $(X,\langle \cdot, \cdot\rangle)$ a Hilbert space and let $A \in B(H)$ a ...
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### Linear operator defined by its eigenvectors/values

Let $H$ be a Hilbert space, $(e_n)$ a complete orthonormal sequence, and $\lambda_n$ a bounded sequence of complex numbers. Let $A$ be defined such that the $(e_n)$ are the eigenvectors of $A$ and the ...
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### Weighted $L_2$ Hilbert space

this is a question where I am trying to find a reference for a result but I haven't been able to find one at all. Define $L_2(\mathbb R,d\mu) = \{g\in \mathbb R: \int g^2d\mu <\infty\}$. I am ...
### Parseval's theorem to $\sum_{k=1}^\infty |\langle x,x_k\rangle|^2$.
Let $\{x_k\}$ be a collection of vectors in a Hilbert space. We take any $x\in H$. The symbol $\langle .,.\rangle$ denote the inner product. The question is as follows. I have to apply the Parseval's ...