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 ...
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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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Multiplication operator is densely defined.

I saw the question Multiplication Operator on $L^2$ is densely defined Disintegration by parts answers using the following argument. If $f \perp \mathcal{D} (L)$ then $\frac{1}{m^2+1} f \in \mathcal{D}...
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What does it mean that a function is unbounded below in every neighborhood?

In this paper Strong Convexity Does Not Imply Radial Unboundedness In [3], Tapia gives this result showing that a strongly convex functional is either radially unbounded (and so minima-existence ...
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Bounded inverse for a closed range closed operator

I am dealing with an unbounded operator $T$ on an Hilbert space $H$. I am interested in proving that it has a bounded inverse $T^{-1}$. I managed to prove that the operator is closed and self-adjoint. ...
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Does this example of symmetric unbounded operators need $\mu$ to be $\sigma$-finite?

From Chapter X. of John B. Conway's textbook A Course in Functional Analysis: 1.10 Example Let $(X,\Omega,\mu)$ be a $\sigma$-finite measure space and let $\phi:X\to\mathbb C$ be a $\Omega$-...
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When is the composition of (unbounded) closed operators closed/closeable?

Let $H_{i}$ be Hilbert spaces for $i=1,2,3$. Let $T_{21}:H_{1} \rightarrow H_{2}$ and $T_{32}: H_{2} \rightarrow H_{3}$ be closed, densely defined, unbounded operators. What are appropriate conditions ...
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Is $TT^*$ (or $T^*T$) densely defined if $T$ is densely defined and symmetric?

Is $TT^*$ (or $T^*T$) densely defined if $T$ is a densely defined and symmetric linear operator? I feel this is untrue, but do you have a counterexample? Thanks
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Compact linear operator definition

It is well known what we mean by a compact linear operator $A:X\to Y$ where $X,Y$ are Banach spaces (see https://en.wikipedia.org/wiki/Compact_operator#Compact_operator_on_Hilbert_spaces). I wonder ...
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Prove the discrete spectrum of $A$ equals to the set of those complex $\lambda$ such that $\lambda I-A$ is Fredholm.

There is considerable divergence in the literature concerning the definition of the essential spectrum of a densely defined closed operator $A$ on a Banach space $X$. I want to find a direct ...
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Why is the maximal domain $D_\text{max}$ of a Sturm-Liouville operator defined the way it is?

Given a Sturm-Liouville type operator which acts on functions on the interval $(a,b)$ $$T:= \frac{1}{w(x)}\left(-\frac{d}{dx}\left[p(x) \frac{d}{dx}\right]+q(x)\right)$$ where $w$, $p^{-1}$ and $q \in ...
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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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Must Linear operators on entire Banach space $l^1$ be bounded?

Assume that it is possible to construct an unbounded linear functional $f$ defined on entire Banach space $l^1$. Then let us consider in unit ball the standard linearly independent sequence $\{e_i\}_{...
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Bounded kernel operator

I have to verify that this operator in $L^2(\Bbb R^3): (Gf)(x)=\int dy f(y) \frac{e^{-|x-y|}}{|x-y|} $ is bounded, using the following rule for the integral kernel: $ sup_x\int dy|a(x,y)|< \infty$ ...
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If $E,F$ are reflexive Banach spaces, then the graphs of unbounded linear operators $A$ and $A''$ are isometrically isomorphic

I'm reading Theorem 3.24 in Brezis's book of Functional Analysis. The statement of the theorem is: Let $E$ and $F$ be two reflexive Banach spaces. Let $ A:D(A) \subseteq E \rightarrow F$ be an ...
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How to prove that the graphs of unbounded linear operators $A$ and $A''$ are isomorphic?

I'm reading Theorem 3.24 in Brezis's book of Functional Analysis. Let $E$ and $F$ be two reflexive Banach spaces. Let $A: D(A) \subseteq E \rightarrow$ $F$ be an unbounded linear operator that is ...
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Proof of involvement

Let $A$ and $B$ be two bounded operators on a Hilbert space $H$. $1)$ $\exists M>0,$ $\forall x\in H$ : $||A^{\ast}x|| \leq M||B^{\ast}x||,$ $2)$ $\exists C$ bounded operator on $H$ : $A=BC.$ We ...
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How to show that the Hilbert-adjoint operator $T^*$ of a (not necessarily bounded) linear operator $T$ on a complex Hilbert space $H$ is bounded

I want to show that if a (not necessarily bounded) linear operator $T$ is defined everywhere on a complex Hilbert space $H$, then its Hilbert-adjoint operator $T^*$ is bounded. Now given that $T$ may ...
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About bounded and closable operators.

Let $T:H\to \mathbb{C}^m$ a densely defined operator with $H$ Hilbert space. Is it true that if $T$ is closable, then $T$ is bounded? For example the differential operator $ T:C^1[0,1]\subset C[0,1]\...
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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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If $A^\star$ is surjective, then there is $c>0$ such that $|u| \leq c |A u|$ for all $u \in D(A)$

I'm trying to prove Theorem 2.21 in Brezis' book of Functional Analysis. The author leaves the proof as an exercise. Could you have a check on my attempt? Let $E, F$ be Banach spaces. Let $A: D(A) \...
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How does $\big ( \|A^{\star} v\| \leq 1 \implies \|v\| \le K \big )$ imply $\exists C, \forall v \in D(A^{\star}):\|v\| \leq C \|A^{\star} v\|$?

I'm reading Theorem 2.20. in Brezis' book of Functional Analysis. Let $A: D(A) \subset E \rightarrow F$ be an unbounded linear operator that is densely defined and closed. The following properties ...
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About the closedness of a certain product of two closed operators?

Let $H$ be a Hilbert space. Let $S∈B(H)$ and let $T$ be a densely defined closed operator such that $TS\subset ST$. Assume further that $T$ is boundedly invertible. Is it true that $ST$ is closed? In ...
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Symmetric (unbounded) operator has a closed symmetric extension

Let $T$ be a symmetric (not necessarily densely defined) (unbounded) operator in a Hilbert space $H$. I want to show, using the theory of the Cayley transform (as developed in e.g. grandpa Rudin), ...
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How are the graphs of $A$ and $A^\star$ related by $I\left[G\left(A^{\star}\right)\right]=G(A)^{\perp}$?

I'm reading Section 2.6 An Introduction to Unbounded Linear Operators in Brezis' book of Functional Analysis Let $E, F$ be Banach spaces and $A:D(A)\subseteq E \to F$ a densely defined unbounded ...
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Is this operator bounded and closed?

Let's define the subset of $\ell^2(\mathbb C)$ $$\mathcal D(A) = \left\{ {z \in {\ell ^2}\left( C \right),\sum\limits_{k = 1}^\infty {k^2{{\left| {{z_k}} \right|}^2} < \infty } } \right\},$$ and ...
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Why is the norm of $x_n(t) = t^n$ for the differentation operator equal to 1?

I am reading Kreyszig's Introduction to Functional Analysis. He explains in an example why the differentiation operator is not a bounded operator. I cannot follow his explanation. I sometimes have ...
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Can the image of a not-bounded "projector" on a normed space be [closed]

Let $X$ be a normed space, and $P:X\to X$ be an idempotent linear map, i.e., $P^2=P$. If $P$ is bounded, then $P(X)$ is closed. Does the converse hold? That is, if $P$ is not bounded, must $P(X)$ not ...
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Must a "projector" on a normed space be bounded?

In V. Moretti's Spectral Theory and Quantum Mechanics, a projector on a normed space $X$ is defined as a bounded linear map $P:X\to X$ such that $P^2=P$. Is the boundedness condition really required ...
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Proving this element-wise multiplication operator in $l^2$ is a closed operator, but not continuous.

I was reading this maths stack exchange post about the difference between closed and continuous linear operators. The example given there is that the operator defined on $l^2$ by $T(x):=(x_1,2x_2,...,...
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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 ...
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1 answer
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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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If $\frac{d^2v}{dx^2}=a^{2}v$ and $v=v_0$ at $x=0$ and $v=0$ at $x=l$, then find the value of $v$

Using $D$-Operator method, in the answer, it is given that $v=v_0\frac{\sinh a(l-x)}{\sinh l}$. But after solving it, i got $v_0=c_1+c_2$ and $c_1e^{al}+c_2e^{-al}$ after solving $m^{2}=a^{2}\implies ...
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second- order parabolic PDE as semigroups ( Evans 7.4.2)

The following is from Evans 7.4.2, second-order parabolic PDE. We assume $L $ has the divergence form, which means $ Lu= -\sum _{i,j=1} ^n (a^{i,j}(x,t)u_{x_i})_{x_j} + \sum_{i=1} ^n b^i(x,t)u_{x_i} +...
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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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Are normal operator with positive self-adjoint part boundedly invertible?

Consider an unbounded normal operator $A$ with domain $\mathcal{D}(A)$ on a Hilbert space $\mathcal{H}$, i.e. its adjoint $A^*$ satisfies $\mathcal{D}(A^*)=\mathcal{D}(A)$ and, for all $x\in\mathcal{D}...
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The operator $Sf=f(1)$ defined on $L^1[0,1]$

Consider the operator $T: L^1[0,1]\to \mathbb{R}$ defined for each $f\in L^1[0,1]$ by $$Tf=\int_0^1 f(y)dy$$ Then it is clear that $T$ is a bounded (linear) operator from $L^1[0,1]$ to $\mathbb{R}$. ...
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Convergence in the resolvent-sense and spectral properties

I'm reading the chapter about unbounded operators in [Reed,Simon,"Methods in modern mathematical physics", vol. 1] Let $\{T_k\}_k, T$ be unbounded selfadjoint operators on a Hilbert space $H$...
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Question 2.23 from Brezis' book of Functional Analysis

Question: Let $E=\ell^{1}$, so that $E^*=\ell^{\infty}$. Consider the operator $T\in \mathcal{L}(E, E)$ defined by $$Tu=\left(\frac{1}{n}u_n\right)_{n\geq 1}$$ for every $u=(u_n)_{n\geq 1}$ in $\ell^1$...
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Question 2.27 from Brezis' book of Functional Analysis

Question: Let E and F be two Banach spaces and let $T\in\mathcal{L}(E,F)$. Assume that $R(T)$ has finite codimension, i.e., there exists a finite-dimensional subspace $X$ of $F$ such that $X+R(T)=F$ ...
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Question 2.24 from Brezis' book of Functional Analysis

Question: Let $E, F,$ and $G$ be three Banach spaces. Let $A:D(A)\subset E\longrightarrow F$ be a densely defined unbounded operator. Let $T\in\mathcal{L}(F,G)$ and consider the operator $B:D(B)\...
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Question 2.22 from Brezis' book of Functional Analysis

Question: $2.22$ The purpose of this exercise is to construct an unbounded operator $A: D(A) \subset$ $E \rightarrow E$ that is densely defined, closed, and such that $\overline{D\left(A^*\right)} \...
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A consequence of the Min-Max Principle for self-adjoint operators

Let $H=(H, (\cdot, \cdot))$ be a Hilbert space. Let $T_1,T_2:D \subset H \longrightarrow H$ be a self-adjoint operators (not necessarily bounded). It's well-know that the spectrum $\sigma(T_i)$ of $...
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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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2 votes
1 answer
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Convergence Criterion in the Domain of an Unbounded Operator

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\...
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