Questions tagged [unbounded-operators]

Let $X$ and $Y$ be normed spaces and $T: D(T)\rightarrow Y$ a linear operator, where $D(T)\subset X$. The operator $T$ is said to be unbounded if there exists a sequence $\{x_n\}\subset D(T)$ s.t. $$\| Tx_n\| \geq n\| x_n\| $$

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"Infinite-dimensional Courant–Fischer"

Let $T$ be a positive self-adjoint operator, possibly unbounded, on a Hilbert space with domain $D$ and spectrum $\sigma(T)$. I know that $$\inf\sigma(T)=\inf\limits_{\substack{x\in D\\\|x\|=1}}\...
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Prove that operator $T_{\alpha}$ is closed

Let $\alpha = (\alpha_n)_{n \in \mathbb{N}}$ be a complex sequence. Define the linear operator $T_{\alpha}$ on $l^2(\mathbb{N})$ by $T_{\alpha}(\varphi_n) = (\alpha_n \varphi_n)$, with domain \begin{...
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Example of an unbounded linear operator which is an open map [closed]

I was going through the open mapping theorem for normed linear spaces and now I can not construct an unbounded linear operator which is an open map. I don't know how to check the openness of a linear ...
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Proving that the limit of $\frac{e^{(tA_\lambda)}u - u}{t}$ as $t\downarrow 0$ exists in a Banach space $X$ for every $u\in X$

Let $X$ be a Banach space and $A$ a closed densely defined (note: not necessarily bounded) linear operator on $X$ and let $\lambda > 0$. Define $$A_\lambda := -\lambda I + \lambda^2 R_\lambda$$ for ...
Epsilon Away's user avatar
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Affiliation of an unbounded operator with a von Neumann algebra

Let $\mathscr{B}([0,+\infty))$ denote the $*$-algebra of all Borel-measurable functions $f: [0, \infty)\to \mathbb{C}$ that are bounded on compact subsets. Given such a function $f$ and an unbounded, ...
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Multiplication of two unbounded operators and functional calculus

Let $A$ be a positive, self-adjoint unbounded operator defined on a Hilbert space $H$. Let $f,g: [0, \infty]\to \mathbb{R}$ be Borel measurable functions that are bounded on compact subsets. We can ...
Andromeda's user avatar
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$A, B,$ and $AB$ densely defined linear operators $\rightarrow (AB)^* \supseteq B^*A^*$

$A, B,$ and $AB$ densely defined linear operators $\rightarrow (AB)^* \supseteq B^*A^*$ This is Exercise 1 from X.1 of Conway's FA. It should be easy, but I'm a bit confused. If $x \in \mathcal{D}(B^*...
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Product of a Hilbert-Schmidt operator and a Kato perturbation of a self-adjoint operator

Let $\mathcal{H}$ be a Hilbert space, $A:\mathcal{H}\rightarrow\mathcal{H}$ a Hilbert-Schmidt operator on it, and $H_0:\mathcal{D}(H_0)\rightarrow\mathcal{H}$ an unbounded self-adjoint operator on it. ...
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Inequality for spectral families of self-adjoint operators

Let $A,B$ be self-adjoint operators on a Hilbert space $\mathcal{H}$ with $D(A) \subset D(B)$ and $A \leq B$. By the spectral theorem, we write : $A = \int_{\mathbb{R}} \lambda \,E^A(d\lambda) \hspace{...
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How is changing the boundary conditions a finite rank perturbation?

I have a question about a statement I came across which I'd be happy to understand more. On $L^2(0,1)$, we can consider two self-adjoint operators. The first operator $H_0$ acts as $H_0f=-f''$, with ...
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The composition of two unbounded linear operator is also unbounded or not?

Let $H$ be a Hilbert space, and consider some linear operators $ A, B: H\rightarrow H $. In functional analysis, I knew that if $A$ and $B$ are both bounded, then the composition $AB$ is bounded. I'm ...
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Spectral family/resolution for $A \otimes 1+ 1 \otimes B$

Let $A, B$ be unbounded self-adjoint operators on Hilbert spaces $\mathcal{H_1}, \mathcal{H_2}$ with spectra $\sigma(A), \sigma(B)$. We know that : $\sigma(A \otimes 1 + 1 \otimes B) = \overline{\...
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Discrete spectrum of $A \otimes 1+ 1 \otimes B$

Let $A, B$ be unbounded self-adjoint operators on Hilbert spaces $\mathcal{H_1}, \mathcal{H_2}$, with both non-empty discrete spectra. Let us say, for instance, $\inf \, \sigma(A) = \lambda_1^A$ and $...
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Banach spaces associated to Sobolev Towers

In the book One Parameter Semigroups for Linear Evolution Equations, the authors provide some definitions as follows: (Page 124). [For $A \in \mathcal{L}(X)$, define] For each $n \in \mathbb{N}, x \...
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Question about inclusion of domains of unbounded operators on a Hilbert space

Assume that T be a self-adjoint operator on the Hilbert space $L^2(0,1)=\{f:\int_0^1 |f(x)|^2 dx<\infty\}$ with the domain $D(T)$ satisfying $C^\infty_c(0,1)\subset D(T)$ (here $C^\infty_c(0,1)$ ...
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Question about domains of extensions of unbounded operators

Let $T$ be a densely defined, symmetric operator on the Hilbert space $\mathcal{H}$ with the domain $D(T)$. Let $T_1$ and $T_2$ be self-adjoint extensions of $T$. Then obviously we have $$ T_1 f=T_2 f ...
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When is $\rho := e^{-\beta H}$ trace-class?

Suppose $H$ is a self-adjoint operator acting on a separable Hilbert space and $H$ has a discrete spectrum with eigenvalues converging to $+\infty$. I want to investigate under what condition the ...
sigma's user avatar
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Proof Position Operator Is Dense

This is an exercise from my last homework sheet, proofing that $P$ is unbounded and self-adjoint was clear, however I'm having trouble proofing that $P$ is densely defined. How my Instructor solved ...
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Domain of sum of momentum operators

Given the tensor product of Hilbert spaces $\otimes_{i \in \mathcal{Z}} (\mathcal{H}_i, \psi_i)$ (here $\mathcal{Z}$ is the set of integer numbers, $\mathcal{H}_i = \mathcal{L}^2(\mathcal{R}, dx)$ and ...
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Show that the image of S under the map $ f \mapsto f * g $ is a compact set in $ C_0([-2, 2])$.

Given $\frac{1}{p} + \frac{1}{q} = 1 $, let $S = f \in L^p(\mathbb{R})$ $spt(f) \subset [-1,1]$, and $\|f\|_p \leq 1$ , and let $ g$ be a fixed but arbitrary function in $ L^1(\mathbb{R})$, with spt(...
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If $f_n\in L^2$, $x f_n \in L^2$, $f_n \to 0$, then $xf_n \to 0$?

Consider the Hilbert space $L^2 = L^2(\mathbb R)$. Consider a sequence $f_n \in L^2$ that satisfies $xf_n\in L^2$. Here, $xf_n$ represents the function $x\mapsto xf_n(x)$. If $f_n \to 0$, then $xf_n \...
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Can one reduce the study of unbounded operators to the one of bounded operators?

So I am not very well-versed in functional analysis, but while studying a problem from theoretical solid-state physics I came across the following question: Suppose $\mathcal{H}$ is a complex Hilbert-...
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Bounded extension of an operator with limited numerical range

In an excercise I'm asked to prove that a densely defined operator, whose numerical range: $$\nu(T)=\{ (\psi,T\psi) \space | \space \psi \in D(T) \wedge ||\psi||=1 \}$$ is a limited subset of $\...
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Brezis' exercise 8.17: the kernel of $A^*$ where $A u=u''-xu'$

Let $I$ be the open interval $(0, 1)$. I'm trying to solve below exercise in Brezis' Functional Analysis, i.e., Exercise 8.17 Let $H=L^2(I)$ and $A: D(A) \subset H \rightarrow H$ be the unbounded ...
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Brezis' exercise 8.17: the domain of $A^*$ where $A u=u''-xu'$

Let $I$ be the open interval $(0, 1)$. I'm trying to solve below exercise in Brezis' Functional Analysis, i.e., Exercise 8.17 Let $H=L^2(I)$ and $A: D(A) \subset H \rightarrow H$ be the unbounded ...
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Bijective Linear Map Construction for continuous and bounded maps: Linear Operator $A$ and $A^{-1}$

Can any one construct a bijective linear map, with $C(E)\simeq C_0(E)$? That is, consider a linear operator $A$, such that $A\colon C(E)\rightarrow C_0(E)$, which can make sense for both $A$ and its ...
Un peti mensonage's user avatar
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Brezis' exercise 8.17: is $D(A^*) \subset H^2 (I)$?

Let $I$ be the open interval $(0, 1)$. I'm trying to solve below exercise in Brezis' Functional Analysis, i.e., Exercise 8.17 Let $H=L^2(I)$ and $A: D(A) \subset H \rightarrow H$ be the unbounded ...
Akira's user avatar
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Brezis' exercise 8.17: the domain of $A^*$ where $A u=u''$

Let $I$ be the open interval $(0, 1)$. I'm trying to solve below exercise in Brezis' Functional Analysis, i.e., Exercise 8.17 Let $H=L^2(I)$ and $A: D(A) \subset H \rightarrow H$ be the unbounded ...
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Existence of an equivalent norm on a Banach space

I am trying to formulated a lemma, because my main result demainds a change of norm. I have an operator $A\colon D(A)\subset X\to X$, $X$ Banach, and $(0,\infty)\subset \rho(A)$. This operator has the ...
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Showing that a graph which is a subspace of $\bigoplus \mathcal{H}_i \oplus \bigoplus \mathcal{H}_i $ is closed.

Let $\mathcal{H}_i$ be a collection of Hilbert spaces, and $T_i \in \mathbf{B}(\mathcal{H}_i)$. Assume that $\sup ||T_i|| = \infty$. Consider $\bigoplus \mathcal{H}_i$(the completetion). Given an ...
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Reed and Simon, Fourier Analysis and Self-Adjointness, second corollary to Theorem X.$25$: how to show that $D(A^2)$ is dense in $D(A)$ for its norm?

This question arose while trying to figure out the proof of the second corollary to Theorem X.$25$ in Reed and Simon's Fourier Analysis, Self-Adjointness, stated as follows: Theorem X.$25$: Let $A : ...
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The Spectrum of the derivative operator in a specific Banach space

Consider the Banach space $X=\left\{u\in C^1([0,1]):\, u(0)=0\right\}$ and the subspace $D=\{u\in C^2([0,1]):\, u(0)=u(1)=u'(0)=0\}$, and the operator $A:D\longrightarrow X$ defined by $Au=u'$. I have ...
amine's user avatar
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Brezis' exercise 8.16.2: determine $R(A)$ and $N(A)$

Let $I$ be the open interval $(0, 1)$. I'm trying to solve below exercise in Brezis' Functional Analysis, i.e., Exercise 8.16 Let $E=L^p(I)$ with $1 \leq p<\infty$. Consider the unbounded operator ...
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Brezis' exercise 8.16.1: check that $D(A)$ is dense in $E$ and that $A$ is closed

Let $I$ be the open interval $(0, 1)$. I'm trying to solve below exercise in Brezis' Functional Analysis, i.e., Exercise 8.16 Let $E=L^p(I)$ with $1 \leq p<\infty$. Consider the unbounded operator ...
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Exponential of operators and commutation

Consider an unbounded self-adjoint and strictly positive operator $A\:\mathcal{D}(A)\to\mathcal{H}$. With strictly positive, I mean $\sigma(A)\subset [a,\infty)$ for some $a>0$. Now, with the ...
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Spectral theory for sum of unbounded, commutative operators

Let's suppose we have two unbounded operators $A$ and $B$ acting on Hilbert space $\mathcal{H}$. In order to simplify everything, let's suppose that $[A, B] = 0$. We have their spectral decomposition $...
MBlrd's user avatar
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How to show $ |(Bx,x)|\leq (Ax,x) $ for any $ x\in D(A) $ here?

On the Hilbert space $ H $, $ A $ is a non-negative self-adjoint operator and $ B $ is a symmetric operator. Let $ D(B)\supset D(A) $, where $ D(A) $ and $ D(B) $ are definite domain for $ A $ and $ B ...
Luis Yanka Annalisc's user avatar
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How to show that $ B $ is bounded with respect to $ A $.

Assume that $ A $ is a densely defined and closed operator and $ B $ is a closable operator. Assume that the definite fields of $ A,B $ denoted by $ D(A) $ and $ D(B) $ satisfies that $ D(A)\subset D(...
Luis Yanka Annalisc's user avatar
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Convergence of self-adjoint operators with converging spectra

Let $A=\int_{\sigma(A)} \lambda dP(\lambda)$ be an unbounded self-adjoint operator on a Hilbert space $H$ with dense domain $D\subset H$. Let $A_n=\int_{\sigma(A)\cap[-n,n]} \lambda dP(\lambda)$ (see ...
ayoo's user avatar
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If $T$ is a symmetric densely defined operator its adjoint $T^*$ is a closed extension of $T$

In this question Why is a densely defined symmetric operator $T$ extended by its adjoint $T^*$? the accepted answer proves that for a densely defined symmetric operator $T$ on a Hilbert space $H$, its ...
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If $T$ is a closable densely defined operator then $T^* = \overline{(T^*)} = T^{***} = (\overline{T})^*$

Let $T$ be a closable densely defined operator on a Hilbert space $H$. In Reed & Simon's book on functional analysis they state the result: $$T^* = \overline{(T^*)} = T^{***} = (\overline{T})^*$$ ...
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Domain of adjoint of annihilation operator

This is Exercise 2.17.1 in What is a Quantum Field Theory by Michel Talagrand. In the notation, inner products are antilinear in the first argument and the complex conjugate is denoted by an asterisk. ...
Buster's user avatar
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If $A$ is maximal monotone and symmetric, then its resolvent and Yosida approximation are symmetric

Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space and $|\cdot|$ its induced norm. Let $A: D(A) \subset H \to H$ be a maximal monotone (unbounded linear) operator. One says that $A$ is ...
Akira's user avatar
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Commutator of self-adjoint operator with semigroup generated by another self-adjoint operator

Let $\mathcal{H}$ be a (complex) Hilbert space. Let $H$ be a self-adjoint operator on $\mathcal{H}$ with dense domain $\mathcal{D}(H) \subset \mathcal{H}$, generating the unitary one-parameter ...
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Domain of adjoint of unbounded finite-rank operators

I am studying closed/closable operators and their adjoints and I am confused: is it true to say that the domain of the adjoint of an unbounded linear functional is necessarily $\{0\}\subset \mathbb{R}$...
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comparison of domains of unbounded operators with complex powers

Let $\cal H$ be a Hilbert space and $T$ be a non-negative invertible self-adjoint (unbounded) operator on $\cal H$. Let $\lambda >0$, and consider $z \in \mathbb{C}$ such that $z=\alpha-i\beta$, ...
DenOfZero's user avatar
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A linear operator is closed if and only if it is weak-weak continuous

Let $X,Y$ be Banach spaces. I want to show that $A: \text{Dom}(A) \subseteq X \rightarrow Y$ is a closed linear operator if and only if for $x_n \in \text{Dom}(A)$ with $x_n \rightharpoonup x$ and $...
Chris's user avatar
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Strictly positive compact operator commuting with a given Fredholm operator

In the following snippet from a paper by Jody Trout on the converse functional calculus, they mention the existence of a strictly positive compact operator $T$ commuting with a given self-adjoint, odd ...
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Can the Stinespring Dilation Theorem extend to *unbounded* operators?

I have some (possibly basic) questions about $C^*$-algebras, the Stinespring Theorem (Theorem 3.6 in Takesaki's book), and unbounded operators. This is motivated by quantum mechanics, where unbounded ...
just a phase's user avatar
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is this (unbounded) operator normal?

I have some question regarding normal operators. I have been given the following definition for the case of bounded linear operators $A\in\mathcal{A}(H)$: $A^*A=AA^*$. My question is: what happens if ...
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