# Questions tagged [unbounded-operators]

The tag has no usage guidance.

80 questions
Filter by
Sorted by
Tagged with
22 views

### What is the analog of the symmetrized Jacobi matrix for delay equations?

For a linear system of ODEs in $\mathbb{R}^{n}$ (with the usual inner product), say $\dot{V}(t) = A(t) V(t)$, we know that if $\xi_{1},\ldots,\xi_{k} \in \mathbb{R}^{n}$ and $V_{j}(t)=V_{j}(t,\xi_{j})$...
75 views

### Spectral representation of closed operators with finite spectral bound

Assume $A$ is a closed linear operator on a Banach space $X$ and is densely defined. Assume the spectral bound $s(A) = \sup\{Re\lambda: \lambda\in \sigma(A)\}$ is finite. For example, if $A$ is the ...
75 views

129 views

### Power of an infinitesimal generator of a $C_0$-semigroup in a Banach space

Let $A$ be the infinitesimal generator of a $C_0$-semigroup of linear operators in a Banach space. Let $n$ be a positive integer, $n\geq2$. Is $A^n$ closed? Here (setting $A^1$ $:=$ $A$, and ...
61 views

### Adjoint for a non-densely defined unbounded operator on a Hilbert space

Let $\mathbf{H}$ be a Hilbert space, and $D$ an unbounded densely-defined operator on $\mathbf{H}$. As is well-known, every such operator admits an adjoint, with domain possibly different from that ...
114 views

### The imaginary exponential of a tangent field on a manifold

If $M$ is a compact Riemannian manifold and $X$ is a tangent field, I am seeking to define the object $\exp {\mathrm i t X}$ for $t \in \mathbb R$, and I do not know how to do it. One option was to ...
158 views

### Spectrum equals eigenvalues for unbounded operator

Let $D$ be an unbounded densely defined operator on a separable Hilbert space $H$. If $D$ is diagonalisable with all eigenvalues having finite multiplicity and growing towards infinity, does it follow ...
97 views

### Non-point spectrum for diagonalisable self-adjoint unbounded operator

Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to$H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...
298 views

374 views

Let $H$ be a separable Hilbert space and let $M$ be a densely defined operator $\mathcal{D}(M) \subset H \to H$. It is closable iff its adjoint $M^{\star}$ is densely defined, and then its closure $\... 1answer 244 views ### Does every integer map generate a von Neumann algebra of type I? Consider a map$m: \mathbb{N} \to \mathbb{N}$(we call it an integer map). Let$E_r$be the set$m^{-1}(\{r\})$. Let$H$be the Hilbert space$\ell^2(\mathbb{N})$and consider the densely defined ... 1answer 220 views ### Are$\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}$and$\| u \|_{W^{2,p}(\Bbb R^d)}$equivalent norms? Is it true that $$\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}\quad\text{and}\quad\| u \|_{W^{2,p}(\Bbb R^d)}$$ are equivalent norms? This results is pretty easy and straightforward for$...
669 views

Let $T$ be an (unbounded) self-adjoint operator. Assume that there is a bounded operator $S$ such that $TS=ST.$ For which kind of $f$ do we have that $f(T)S=Sf(T)?$ My thought was that using a ...
112 views

### Bounded self-adjoint perturbation of a p-summable spectral triple

I am new to the field of Noncommutative Geometry.I was reading the chapter on Spectral triple from the book 'Elements of Noncommutative Geometry' by Gracia-Bondía,Várilly and Figueroa.Now,after ...
141 views

### Symmetric diagonalizable operators and self-adjointness

Given a densely defined symmetric operator $L$ on a Hilbert space $H$, which is also assumed to be diagonalizable, will there always exist a unique extension of $L$ to a self-adjoint operator?
249 views

### Minimum eigenvalue of One-dimensional Schrodinger Operator

Consider the One dimensional Schrodinger Operator $$-\frac{d^2}{dx^2} + V(x)$$ Where the Potential Function $V$ is of the form $V(x) = x^4 - a^2x^2$ , $a \in \mathbb{R}$. Now of course,the ...
223 views

### Spectral growth of One dimensional Schrodinger Operator

Conside the One dimentional Schrodinger Operator $$-\frac{d^2}{dx^2} + ( V(x) + E )$$ Where the Potential Function $V$ is of the form $V(x) = ax^2 + b^2x^4$ , $a,b \in \mathbb{R}$. What is known ...
643 views

### On the domains and extensions of unbounded operators

I am not an expert in functional analysis but I was studying some, motivated from some mathematical physics considerations. I am not quite sure whether this is research-level, but let me state some ...
140 views

### Scattering of relativistic particle by long-range potential

Let $\mathcal{H}=L^2(\mathbb{R}^3)$, $H_0=\sqrt{-\Delta+M^2}$, ($M$ is a positive constant, $\Delta$ is the laplacian) and $H=H_0+V(\vec{x})$ (where $V(\vec{x})$ is the operator of ...
358 views

### Notations for dual spaces and dual operators

I'm asking for opinions about the 'best' notations for: 1. the algebraic dual of a vector space $X$; 2. the continuous dual of a TVS; 3. the algebraic dual (transpose) of an operator $T$ between ...
230 views

### Is the range of a weak convergent sequence also a weak convergent sequence?

Under which condition the image of a weakly convergent sequence (xn) by an unbounded linear operator T is a weakly convergent sequence (Txn) ?
226 views

### Inverse and implicit function theorems with domain

I have seen an author use the Implicit Function Theorem for a map whose second partial derivative has a bounded inverse, but is unbounded. The map itself is not defined on an open set, but only on a ...
156 views

### Stochastic integral is a continous or closed operator?

The Setup Let $\xi_t$ be a process adapted to the filtration $\mathfrak{F_t}$ of the semi-martinagale $X_t$, such that both are square integrable. Then is the map \begin{align} F_T: L^2(\mathfrak{...
209 views

### Selfadjointness of hamiltonian with 1/x potential

Let us consider the Hilbert space $L^2([0,\infty))$ and operator $H=-\frac{d^2}{dx^2} + \frac{1}{x}$ on the domain of $C^{\infty}_0((0,\infty))$ (smooth functions with compact support away from $0$). ...
58 views

### On extending a semigroup property

Let $T(t)$ be a $C_0$-semigroup on a Hilbert space $H$ with a generator $A$. It is well known that for all $x\in H,$ we have: $\int_0^t T(s)x ds \in D(A)$ and $A\int_0^t T(s)x ds = T(t)x-x$. How ...
51 views

### Simplify the expression of $T^+$ for an unbounded operator $T$?

For a negative unbounded operator $T$, what equals the operator $$T^+ = \left[\frac{1}{2}(|T| + T) \right]^{**},$$ where $|T|= (T^2)^{1/2}$ and $A^{**}$ is the minimal closed extension of an ...
115 views

### For self-adjoint $T$ on $L^2(\mathbb{R}^n)$, when does $(1 + |x|)^{-1} (T - i \varepsilon)^{-1}(1 + |x|)^{-1}$ have a limit as $\varepsilon \to 0$?

Let $T : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$ be a possibly unbounded self adjoint operator. Let $R_\varepsilon$ denote the resolvent $(T - i \varepsilon)^{-1}$, $\varepsilon > 0$. Suppose that,...
164 views

### Unbounded operator [closed]

Is the Laplacian operator $\Delta: C^2([a,b],\mathbb{R}) \to L^2(a,b)$ unbounded? Here $a,b \in\mathbb{R}$, $C^2([a,b],\mathbb{R})$ associated with $L^2$ norm. If yes, how can we "modify" these ...
I'm asking a question about Lie group representation. Let $G$ be a Lie group, not necessarily connected. Let $\Omega$ be an element in the center of the universal enveloping algebra $U(\mathfrak{g})$ ...