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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Kesavan's book example 4.2 ; adjoint operator for banach spaces

Let V and W be Banach spaces and A:D(A)-->W a densely defined linear operator. If G(A) graph norm is dense in VxW , then show that D(A^*) ={0}. I know every adjoint operator contains zero but ı ...
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Does $(u \delta u^{*})^{it} = u \delta^{it} u^{*}$ hold, where $\delta$ is a (positive) self-adjoint unbounded operator and $u$ unitary?

Let $\delta$ be a positive self-adjoint unbounded operator and let $u$ be a unitary operator on an infinite-dimensional Hilbert space. Is it true that $(u \delta u)^{it} = u \delta^{it} u^{*}$ for ...
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Prove the operator defined by $Lf=x^2f(x)$ is not bounded.

Let the operator $L: dom(L) \subset L^2(\mathbb{R} ) \to L^2 (\mathbb{R})$ defined by $Lf(x)=x^2f(x)$ where $dom(L)=\{ f \in L^2(\mathbb{R} ) : x^2 f(x) \in L^2 (\mathbb{R}) \}$. Prove that $L$ is ...
1 vote
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Sum of adjoints of two operators is a subset of adjoint of sums.

Let $T : \mathcal{D} (T) \subset H \to H$ and $A: \mathcal{D} (A) \subset H \to H$ two densely defined operators and $H$ a complex Hilbert space. Prove $T^*+A^* \subseteq (T+A)^*$. My solution is as ...
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Action of an operator-valued function of the momentum operator $\hat{p}$ and its unboundedness

I am currently dealing with an operator-valued function $f(\hat{T})$ of the following kind: $$f(\hat{T}) =\sqrt{1 + b\hat{T}^2}$$ where $b$ $\in \mathbb{R}$ and $\hat{T}$ is a linear operator acting ...
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How to prove T is a closable operator

Let $T:H\to H$ a densely defined operator, with $H$ a Hilbert space such that: $$Re(x,Tx)\geq 0, \forall x\in Dom(T)$$ I want to prove that $T$ is a closable operator, that means... that there exists ...
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Define a unbounded functional on C00 ⊂ l 2 . Justify your answer

I know that Not possessing both an upper and a lower bound. So for all positive real values V there is a value of the independent variable x for which |f(x)|>V. For example, f (x)=x 2 is unbounded ...
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How can I prove that this Linear operator is closed?

I know that for every infinite dimensional Banach space $X$ admits a linear and discontinuous operator. To prove it, I took a sequence $\{e_n\}_{n\geq1}$ of linearly independent vectors of $X$, then I ...
1 vote
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Hahn-Banach Theorem and non bounded linear functionals.

I have read the Hahn-Banach theorem, which states de following: Let $X$ be a real vector space and $p$ a sublinear functional on $X$. Furthermore, let $f$ be a linear functional which is ...
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Adjoint of the direct sum of operators vs. direct sum of their adjoints

Let $\mathcal{H}$ be an infinite-dimensional complex Hilbert space which decomposes as the direct sum of a countable family of Hilbert spaces $\{\mathcal{H}_n\}_{n\in\mathbb{N}}$, namely \begin{...
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Example for trivial intersection of domains

For a bounded operator $A$ in a Hilbert space $\mathcal H$ the real part of $A$ is defined by $\operatorname{Re}(A) = \frac 12(A+A^*)$. However, if $A$ is unbounded, this operator is defined on the ...
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Preadjoint of unbounded operators on Banach spaces

Let $X$ and $Y$ be two Banach spaces. Let $T : Y^* \rightarrow X^*$ be a unbounded linear map where $X^*$ represents the Banach dual of $X$. My question is, there exists a dense domain $D \subseteq X$ ...
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Proof of an unbounded operator in sequence space

Let $\ell^2=\ell^2(\mathbf C)$ (with $\mathbf C$ the complex numbers) be the sequence space over the complex numbers with the canonical Hilbert basis $(e(j))_{j\in\mathbf N}$ ($e(j)=\delta_{jk}$) and ...
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My question is somewhat close to this one, but the counterexamples given there do not apply here. Setup. Given a Hilbert space $\mathcal H$, a closed operator $A$ and a convergent sequence \$(x_n)_{n\...