# 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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### Prove that the formula correctly defines an operator on a Banach space

Let $A$ be a bounded operator on a Banach space. Prove that the formula $e^A = \sum_{k=0}^\infty \frac{A^k}{k!}$ correctly defines operator $e^A$. Prove also that $\|e^A\| \le e^{\|A\|}$. I suppose ...
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### On positive self adjoint unbounded operator

This is a theorem from Rudin Functional Analysis. $T$ is a self adjoint operator in $H$ (a Hilbert space). $T$ is not necessarily bounded and $\mathscr D(T)$ denotes the domain of $T$. We are to show ...
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### $T$ unbounded operator not closed $\implies$ resolvent of T is empty?

I am studying the subject of unbounded operators and I'm wondering why if an operator is not closed than his resolvent is empty. Thanks !
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### Green's functions in weak formulations: unbounded linear form

Consider a weak formulation of some PDE: $$\text{Find } u\in H^1_0(\Omega)\text{ s.t.:}\qquad\qquad\\ B(u,w) = L(w)\quad \forall\, w\in H^1_0(\Omega),$$ with $H^1_0(\Omega)$ the typical first order ...
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### Unbounded operator spectrum theorem

$\textbf{Thm}$Let $A$ be an unbounded self-adjoint operator on a separable Hilbert space $H$ with domain $D(A)$. There there exists a measure sapce $(M,\mu)$ with $\mu$ a finite measure, a unitary ...
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### Is the composition of closed unbounded operators closed?

Let $H$ be a complex Hilbert space with linear subspaces $U,V$. Then a (not necessarily bounded) linear operator $T:U\to V$ is said to be closed if $$\text{Graph}(T)\equiv\lbrace(u,Tu):u\in U\rbrace$$ ...
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### condition for postive symmetric operator to be essentially self-adjoint

Suppose $A$ is a densely defined symmetric operator on Hilbert space which is positive. (a) prove $||(A+I)\phi||^2\geq ||\phi||^2+||A\phi||^2$ (b) Show $Ran(A+I)$ is closed if $A$ is a closed operator ...
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### adjoint of unbounded differential operator

Consider $T=-\frac{d^2}{dx^2}$ as an operator on $L^2(\mathbb{R})$ with domain $C_0^\infty(\mathbb{R})$. What's the adjoint of T? This is an exercise from Reed&Simon's book. I have managed to ...