# 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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• 45
34 views

### What does strong resolvent convergence tell about spectrum of the limit?

I have a sequence of operators $A_n$ on a separable Hilbert space (not necessarily with the same domain). These operators are unbounded, self-adjoint, and converge in the strong resolvent sense to ...
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### On the domain of unbounded self-adjoint linear operators on Hilbert spaces

Suppose I have a linear unbounded operator $A:\operatorname{dom}(A) \to \mathscr{H}$, with $\operatorname{dom}(A)$ carefully chosen to make $A$ self-adjoint ($A^\dagger=A$). Could $A^n$ be ...
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### Show that the set is not bounded - formal proof - $D = \{(x,y)\in R^2|y=\frac 1 x\}$

is there any way to prove with epsilon \ other way that the set is unbounded? $$D=\{(x,y)\in R^2\mid xy=1\} = \{(x,y)\in R^2\mid y=\frac 1 x\}$$ Here is a proof my exerciser at class did ( which is ...
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• 1,940
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### Factorization of an unbounded operator as composition of a bounded operator composed with one whose inverse is bounded

Does every densely-defined unbounded operator $P$ have a factorization'' of the form $$P = A B^{-1},$$ with $A$ and $B$ bounded? Given two bounded operators $A,B$ such that $B^{-1}$ is a densely-...
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### Construct an Injective and onto unbounded operator.

i was study functional analysis and i found a interesting problem. Let $X$ an infinite dimensional normed space. Construct an operator $T: X \rightarrow X$ such that $T$ is injective and onto. Also, ...
• 61
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### Is this linear map continuous?

Let $A:L^{2}\left( (0,1)\right) \rightarrow L^{2}\left( (0,1)\right)$ be defined as $Af\left( x\right) =-\ln \left( x\right) f\left( \frac{x}{2}% \right)$ for every $f\in L^{2}\left( (0,1)\right)$ ...
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### Adjoint of a Densely Defined Unbounded Operator is Unique

Let $\mathcal H$ be a Hilbert space and $A: \mathcal D_A \to \mathcal H$ be an unbounded linear operator. Suppose also that $\mathcal D_A$ is dense in $\mathcal H$. We define the adjoint of $A$, $A^*$ ...
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### The domain of the cube operator.

Let $E$ be a Banach space, and $A$ an unbounded operator with domain $D(A)$, we assume that $D(A)$ is dense in $E$ and $A$ is closed (i.e its graph is closed) and $\rho(A)$ (the resolvent set of $A$) ...
• 177
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### Venn diagram for basic types of operators

This Venn diagram is an attempt to visually classify densely-defined linear operators between Banach spaces (self-adjoint operators are an exception, defined between Hilbert spaces). Operators are ...
• 437
1 vote
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### Prove this Fourier series converges to a continuous function

Problem Consider the space $\mathcal D$ of continuously differentiable functions on the unit circle and the operator on $L^2$ (of the unit circle) with domain $\mathcal D$ given for $f\in\mathcal D$ ...
• 215
1 vote
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### Unbounded closed operator

Let $E,F$ be two Banach spaces and $A$ an unbouded operator from $D(A) \subset E$ to $F$. We want to prove the following lemma : Let $A$ be a closed operator. The following are equivalent : $A$ is ...
• 177
1 vote
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### What are simple conditions for the adjoint of a positive, unbounded, densely defined operator on a Hilbert space to be positive?

I'm reasking this deleted question because I believe I've made a some progress towards an answer, which I'm also interested in knowing. Here's the restatement: Suppose $\ A\$ is a densely defined, [...
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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 , 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. ...
1 vote
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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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• 425
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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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1 vote
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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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