# Tagged Questions

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

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### Show an operator is compact if $\sum \|Te_n\| < \infty$

Let $H$ be a separable Hilbert space, define a bounded linear operator $T:H \rightarrow H$, show it is compact if $\sum \|Te_n\|_H < \infty$. My attempt: We show that $T(B)$ is totally bounded....
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### Non-separable Hilbert spaces in duals

A topological space $X$ satisfies the countable chain condition if every family of pairwise disjoint open sets in $X$ is countable. I am looking for a reference to the following fact: Suppose that $X$...
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Let $y\in c_0$ and define the operator from $l^2 \rightarrow l^2$ as the following $$T\bigg(\sum x_n e_n\bigg) \mapsto \sum y_n x_n e_n.$$ I have shown that the operator is continuous, compact and ...
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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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### A matrix as eigenvalue?

I wonder if some work has been developed on operators in Hilbert space that have the property of having matrices instead of numbers as eigenvalues (the matrices do not necessarily act on vectors in ...
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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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Given $f\in L^2(\mathbb R)$ and $g(x)=\int^{x+1}_x f(t)\,dt$ (a) Relate $\hat f$ and $\hat g$ (b) Prove $g\in H^1(\mathbb R)$ Part A: $$\hat g(k)= \int^{\infty}_{-\infty} \int^{x+1}_{x} f(t)\,... 1answer 89 views ### Chain of closed subspaces in a Hilbert Space Let H be a separable Hilbert Space (WLOG, we may assume H=\ell_2(\mathbb{N}) is the space of square summable sequences). Can there exist an uncountable chain of closed subspaces? In other words, ... 1answer 57 views ### If a compact operator satisfies T^nx\to0 weakly for all x, then \|T^n\|\to0 Let H be a real Hilbert space, T:H\to H be a compact operator. Suppose that for every x\in H, sequence (T^n x)_{n\in \mathbb{N}} converges weakly to 0. How to prove that  \lim_{... 1answer 108 views ### Does S^\bot+T^\bot = (S\cap T)^\bot hold in infinite-dimensional spaces? If S and T are subspaces of some finite-dimensional inner product space then$$S^\bot+T^\bot = (S\cap T)^\bot.$$See, for example, this post or this post Does it hold in infinite-dimensional ... 1answer 3k views ### The difference between an isometric operator and a unitary operator on a Hilbert space? It seems that both isometric and unitary operators on a Hilbert space have the following property: U^*U = I (U is an operator and I is the identity operator) What is the difference between ... 1answer 73 views ### Role of metric in the matrix representation of Hermitian adjoint I'm working through Jeevanjee's "An Introduction to Tensors and Group Theory for Physicists", and while trying to prove that the matrix representation M(A^\dagger) of a Hermitian adjoint A^\dagger ... 0answers 121 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 ... 2answers 316 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 401 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 26 views ### Proving a particular subset of a Hilbert space is a subspace I have a small question please, how to prove that this set: F=\lbrace h\in H, \langle f''(u)h,h\rangle <0\rbrace is a sub space of the Hilbert space H, where f''(u) is a self-adjoint ... 0answers 40 views ### Spectral decomposition of a Hilbert space I have this proof, but I don't understand how they do the spectral decomposition of H into H_-and H_+? Please help me. Thank you. 1answer 219 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\}.$$Then we say that X ... 1answer 46 views ### Prove that a linear and continuous operator admits inverse in Hilbert space Let (H,(\cdot,\cdot)) an Hilbert space and A:H\rightarrow H a linear and continuous operator such that there exists \alpha >0 such that$$(Au,u)\geq \alpha \|u\|^2 \text{ for each } u\in H.$$... 1answer 154 views ### Skew-adjoint differential operator B with spectrum \sigma(B)=i(-\infty,-1] Consider the Hilbert space X=L^{2}\left(\mathbb{R}^n\right) and the Schrödinger operator A=i\Delta defined on the domain D(A)=H^2(\mathbb{R}^n). It is known that the spectrum of A is \sigma(A)... 1answer 989 views ### Convergence: Weak vs. Strong Given a Hilbert space \mathcal{H}. Suppose one has:$$\|\varphi\|=\lim_n\|\varphi_n\|$$Then it follows:$$\varphi\rightharpoonup\varphi\implies\varphi_n\to\varphi$$How can I check this? 1answer 606 views ### Square Summable functions Can somebody please help me understand the notion of square summable functions intuitively?? I have been self studying Hilbert Spaces and Fourier Transform for DSP. Any help is appreciated. Thanks. 0answers 64 views ### Prove that operator is completely continuous Let's consider Banach space \ell^\infty of bounded sequences x = \{ \xi_n\}_{n=1}^\infty:$$ ||x|| = \sup_{n \in \mathbb N} |\xi_n|. $$Suppose matrix ||a_{i j}||_1^\infty specifies operator A ... 1answer 226 views ### Weakly sequentially continuous operators in Hilbert space are norm continuous. Suppose I have a linear operator T from a Hilbert space H to itself, and T maps every weak convergent sequence to a weak convergent sequence. Show that T is continuous. I feel that this statement ... 1answer 146 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 65 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 \sigma_{\epsilon}(T)\... 1answer 137 views ### How to find all isometries of Hilbert space? We know all isometries of \mathbb R^n  are composition of transfer by orthogonal linear functions. How to find all surjective isometries of Hilbert space? Is there similarity? 0answers 89 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 inner ... 1answer 90 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 97 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 adjoint.... 1answer 72 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 u\left(t,x\right)-mu\left(t,x\right)+h\...
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 ...