# 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. $$\| Tx_n\| \geq n\| x_n\|$$

410 questions
Filter by
Sorted by
Tagged with
38 views

• 772
57 views

### 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 ...
• 2,860
56 views

### 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 ...
• 337
117 views

### 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 ...
• 175
59 views

### 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(...
• 1,486
1 vote
55 views

28 views

### 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 ...
• 17.4k
1 vote
36 views

### 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 ...
• 17.4k
59 views

### 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 ...
94 views

### 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 ...
• 17.4k
33 views

### 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 ...
• 17.4k
1 vote
81 views

### 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 ...
• 549
80 views

### 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 ...
• 559
174 views

• 175
70 views

• 1,036
33 views

### 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 ...
• 11
53 views

### 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 ...
• 5,965
71 views

### 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: $$T^* = \overline{(T^*)} = T^{***} = (\overline{T})^*$$ ...
• 5,965
88 views

### 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. ...
• 235
1 vote
61 views

### 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 ...
• 17.4k
45 views

### 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 ...
• 125
1 vote
60 views

### 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 $\{0\}\subset \mathbb{R}$...
• 460
41 views

### 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 $z=\alpha-i\beta$, ...
• 125