Operator theory is the branch of functional analysis that focuses on bounded linear operators, but it includes closed operators and nonlinear operators. Operator theory is also concerned with the study of algebras of operators.

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A separating set which is not cyclic

Let $H=L^2[0,1]$ , $T_g$ be the multiplication operator on $H$, i.e. $f\to fg$ . Let $A$ be the set of the $T_g$ as $g$ runs through the set of polynomials with complex coefficients. Let $h$ be te ...
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0answers
44 views

Strong and weak equivalence of $C^*$-extensions by compacts

Let $A$ be a $C^*$-algebra. An extension of $A$ by the compact operators $K$ is an embedding $\epsilon$ of $A$ into the Calkin algebra $B(H)/K$. Two embeddings $\epsilon_1$ and $\epsilon_2$ are ...
0
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1answer
41 views

Inequality on Hilbert spaces in order to prove the nonexpansivity of a mapping.

I have an application $T\colon H\to H$ (where $H$ is a Hilbert space) such that $$(Tx-Ty,x-y)\leq \|x-y\|^2,\forall x,y\in H$$ where $(\cdot,\cdot)$ is the inner product of $H$ and $\|\cdot\|$ its ...
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1answer
70 views

Positive semidefinite linear operator $T$ satisfies $T^k=I$. Is $T$ the identity?

I got the following question in an exam I got yesterday that I didn't managed to answer: Let $V$ be a finite dimensional unitary vector space and let $T:V \to V$ be a positive semidefinite linear ...
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1answer
77 views

boundary of a spectrum proof

Let $A$ be a closed unital subalgebra of banach algebra $B$. Prove that ${\delta}{\sigma}_{B}(x)$ is contained in ${\delta}{\sigma}_{A}(x)$ for every $x$ in $B$.
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1answer
26 views

Can we find an invertible projection in an arbitrary von Neumann algebra?

I am looking for an answer for this question: Let $\mathcal{A}$ be an arbitrary von Neumann algebra, can we say there is an invertible projection ($P\neq I$) in $\mathcal{A}$? I think, if there is ...
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0answers
162 views

renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
0
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1answer
72 views

homomorphism or not

Let $T$ be a bounded operator on $H$ and fix a vector $x\in H$. Define $f$ on the space of polynomials in $T$ by $f(p(T))=p(x)$. Is $f$ a homomorphism? Initally I thought it obvious but the subtelty ...
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0answers
85 views

Examples of operator theory on Hilbert space

$(1)$ If $T \in B(H)$ is self-adjoint and $T \neq 0$ then $T^n \neq 0$ $(a)$for $n=2,4,8,16,... (b)$ for every $n$ $(2)$ Show that any $T \in B(H)$ can be uniquely expressed as $T=T_1+iT_2$ ...
6
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1answer
86 views

Can we say $TT^{*}=T^{2}$ implies $T=T^{*}$?

Let $A$ be a $C^{*}$-algebra, Can we say $TT^{*}=T^{2}$ implies $T^{*}=T$? for $T\in A$ I am looking for a counterexample! Thanks
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1answer
61 views

Surjective homomorphism example

What is an example of a surjective homomorphism $B(H)\to\mathbb C$, where $B(H)$ is the set of bounded linear operators on a Hilbert space $H$, and $\mathbb C$ is the complex numbers.
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1answer
117 views

Behavior of the resolvent near the boundary of the spectrum

My question is, in some sense, a continuation of the question below. Isolated singularities of the resolvent Suppose $T\in B(H)$ has no eigenvalues, pick $x\in H$, $x\neq 0$, and consider the ...
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0answers
44 views

How to find all the eigenvalues of a positive operator whose eigenvectors are positive semi-defintie?

A linear operator $T:\mathcal{H}_n\rightarrow \mathcal{H}_n$ is said to be positive if $T(\mathcal{P}_n)\subset\mathcal{P}_n$ where $P_n$ is the set of positive semi-definite matrices. For a positive ...
1
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1answer
50 views

Gelfand transform explicity

Let $T$ be a bounded normal operator. Let $A$ be the algebra generated by $T$ and $T^*$. What is the explicit Gelfand transform $G:A\to C(\sigma(T))$? My book says the image of $T$ is the ...
0
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1answer
37 views

How to prove that $A B A^* \leq \|B\| A A^*$ for operators A,B?

Let $A$, $B$ bounded operators on a Hilbert space $H$. Further let $B$ be self-adjoint. Then we have that $A B A^* \leq \|B\| A A^*$. I wanted to ask how to prove this inequality or where I can find ...
0
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1answer
61 views

Closed graph theorem question?

Let $H$ be a Hilbert space. Let $A:\operatorname{dom}A\to H$ has a closed graph, where $\operatorname{dom}A$ is dense in $H$. Let $S\subseteq \operatorname{dom}A$ be dense. Is it true $A_{|S}$ has a ...
0
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1answer
35 views

Gelfand Transform in a specific case

What is the gelfand transform of an operator in the algebra generated by a bounded normal operator and it's adjoint? Thanks
2
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1answer
53 views

Unbounded extension of bounded operator

Is it possible to construct an unbounded extension of bounded densely defined operator? To be more concrete, let $\mathcal{H}$ be Hilbert space, $\mathcal{D}\subset\mathcal{H}$ - a dense subset, ...
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5answers
266 views

Integral representation for $\log$ of operator

How can one prove that $$ (\log\det\cal A=) \operatorname{Tr} \log \cal{A} = \int_{\epsilon}^\infty \frac{\mathrm{d}s}{s} \operatorname{Tr} e^{-s \mathcal{A}},$$ for a sufficiently well-behaved ...
1
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1answer
46 views

$ \|T_n\| \not \rightarrow \|T\|$ even if $ T_nx \rightarrow Tx, \forall x $

There is a theorem which states that given $X$ normed space, $Y$ Banach space on $\mathbb R, D \subseteq X$ dense and $T_n \in \mathcal L(X,Y)$ a bounded sequence s.t. $T_nz$ converges $\forall z \in ...
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0answers
51 views

Operators on a Hilbert space question

For a Borel measure $\mu$ define $\langle S_\mu x,y\rangle=\int_H\langle x,z\rangle \langle y,z\rangle \mu(z)$. An exercise in my book that I am reading says that I could find a $\mu$ s.t. $S_\mu$ ...
3
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2answers
109 views

Isolated singularities of the resolvent

Let $T$ be a bounded operator on $l_2$ such that there exists $\mu$ in the spectrum of $T$ which is an isolated point of the spectrum. We know that for any $x\in l_2$ the resolvent function ...
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1answer
45 views

$T:V\rightarrow V $ is over $\mathbb{R}$ , it's matrix is $A$, $A=PDP^*$. Is it true that $A$, $D$, and $P$ are in $M_{n \times n}(\mathbb{R})$

$T:V\rightarrow V$ is over $\mathbb{R}$ and $V$ of finite dimension $n$, and I know that it is orthogonally diagonalizable. The Matrix that represents it - call it $A$ ,in orthonormal basis is ...
2
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2answers
433 views

$T^*T=TT^*$ and $T^2=T$. Prove $T$ is self adjoint: $T=T^*$ [duplicate]

$V$ is an inner product space of finite dimension over $\mathbb{R}$, and $T:V\to V$ a linear transformation which is normal, that is, $T^*T=TT^*$. In addition $T^2=T$. Prove $T$ is self adjoint, that ...
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2answers
42 views

Adjoint of an operator question.

Let T be a normal operator. Prove that $\|T\|^{2n}=\|TT^*\|^n$ Has it got something to do with $\|T\|=\|T^*\|$?
4
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1answer
79 views

Isometry between $l^p$ and $L^p$.

Consider $p\in[1,\infty)$ and the operator $T:l^p\rightarrow L^p([0,\infty))$: $$ Tx=\sum_{n=1}^\infty x_n\chi_{[n-1,n]} \qquad\forall\,x\in(x_1,x_2,\ldots,)\in l^p $$ Prove that $T$ is an isometry. ...
3
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0answers
61 views

operator on separable banach space whose spectrum and point spectrum is prescribed compact set

I am interested in obtaining the following paper: G. K. Kalisch, "On operators with large point spectrum," Scripta Math. 29 No. 3-4, (1973), 371-378. According to Ben Mathes, "Strictly Cyclic ...
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0answers
36 views

can we prove that a certain supremum of affine functions is frechet differentiable or at least continuous?

Let $X$ be a Hilbert space. Let $A\colon \operatorname{dom} A\to X$ be linear operator with closed graph (not necessarily bounded). Define $$ g\colon \operatorname{dom}A\to \mathbb{R}:x\mapsto ...
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1answer
64 views

Spectral decomposition

For a compact normal operator, the space can be written as the sum of generalized eigenspaces. So every element can be written as a linear combination of the eigenvectors, one from each eigenspace. ...
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2answers
142 views

If $T^2=TT^*$ then can i conclude that $T=T^*$?

let $B(H)$ be all bounded operator on Hilbert space H. If $T^2=TT^*$ then can i conclude that $T=T^*$? I think this is true if T is one to one. Can i construct an example that shows it is not true for ...
2
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0answers
51 views

spectral projections

Let $A$ be a von Neumann algebra, and $T$ be an hermitian element of $A$. Show that the spectral projections of $ T $ belong to $A$. Proof: the spectral projections of $ T $ commute with every ...
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1answer
45 views

Abelian von Neumann algebra

Let $M$ be a family of commutative normal operators which is closed under adjoint. clearly $M\subset M'$, but I do not know why $M'\subset M^{''}$? and how can conclude that $M^{''}$ is abelian?
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1answer
283 views

Convergence in the strong/weak operator topology: nets versus sequences

Let $H$ be a separable infinite dimensional Hilbert space, with orthonormal basis $(e_n)_{n=1}^\infty$. Consider the operators $U_n$ on $H$ such that $U_ne_k = e_1$ if $n=k$ and $U_ne_k=0$ if $n\neq ...
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0answers
52 views

Derivative on a function of tensor products

Assume I have defined an operator $A \otimes B$ on a $H \otimes L^2(\mathbb R^d)$ where $H$ is a Hilbert space as in Reed/Simon p. 299. $A$ is an operator on $H$ and $B$ is an operator on $L^2(\mathbb ...
0
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1answer
32 views

sequence of closed subspaces is not strictly increasing

Let H be an infinite dimensional hilbert space and $A_{n}$ a sequence of closed subspaces. Prove the sequence is not strictly increasing. I suppose it's sort of intuitive but what is a formal proof?
1
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1answer
131 views

If A unitary matrix and orthogonally diagonalizable why there is a basis in whichthe linear trans. matrix is diagonal?

If $A$ is a $n\times n$ unitary matrix (above the complex field) and is orthogonally diagonalizable, why does it mean that the is an orthonormal basis $\mathbb C$ in which the matrix that represent ...
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2answers
61 views

Operator Theory Textbook Question

I read the following excerpt in my course textbook: Now, I'm led to believe that $P:X\rightarrow X$ as above is bounded iff $M$ and $N$ are both closed. I understand the only if direction, but I ...
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1answer
26 views

Positive elements below projections

Let $a$ be a positive element in $A$, where $A$ is a $C^*$-algebra. Let $p\in A$ a projection and suppose $a\leq p$. Is it true that $ap=pa$? If yes, shouldn't we have $ap=pa=a$, since ...
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3answers
177 views

Exercise about compact operator.

In $X=\ell^p$, $p\in[1,\infty]$ we consider: $$ T(x_1,x_2,x_3,\ldots)=(0,x_1,0,x_3,\ldots) $$ Prove that $T$ isn't a compact operator and that $T^2$ is a compact operator. I think I solved the second ...
3
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3answers
148 views

Reference for a Proof of Weyl-Von-Neumann Theorem

I'm looking for a reference for the proof of the Weyl Von Neumann theorem, however there seems to be two (or the two might be the same). There's the one which is stated in Conways, A Course in ...
3
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1answer
173 views

Bounded Inverse Theorem

$A$ is a bounded linear operator from $X$ to $Y$ (both Banach spaces). Show that if there exists $k > 0$ such that $\|Ax\| \geq k\|x\|$, for all $x$ then $\operatorname{range}(A)\,$ is closed. My ...
4
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2answers
363 views

Gateaux and Frechet derivatives and related notions

Let $X$ and $Y$ be normed real vector spaces, and $f : X \to Y$ a map. Let's say that: G) $f$ is Gateaux differentiable at $x_0 \in X$ if for all directions $v \in X$ the limit $f'(x_0)(v) := ...
0
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1answer
28 views

unbounded operator with no invariant subspace

Is there an example of an unbounded operator on a hilbert space with no invariant subspace? Or is there some other reason why we are only interested in bounded case.
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1answer
56 views

Condition for vector to be in the domain of unbounded operator.

Let $P$ be unbounded self-adjoint operator on some Hilbert space $\mathcal{H}$. We assume that the limit $$ \lim_{\epsilon \searrow 0} \|\exp(-\epsilon^2 P^2/2) P\psi\| $$ exists and is finite. Does ...
0
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2answers
99 views

Spectral Theorem for normal operators

I want to prove this in the infinite dimensional Hilbert space case. What is the easiest way to go about this (What do I need to know, what theorems do I need,etc). My aim is to show every normal ...
0
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1answer
195 views

Commutating operators and tensor products

I have this lecture slides about commutators and tensor products, but there is one part that I don't understand: The operators and are commuting operators on the tensor product and their sum has ...
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1answer
56 views

Exercise about linear operator

For $X$ Banach, I have to show that if $T\in\mathfrak{L}(X)$ and $||T||_{\mathfrak{L}(X)}<1$ then exists $(I-T)^{-1}$ and $$ (I-T)^{-1}=\sum_{n=0}^\infty T^n. $$ For the existence of $(I-T)^{-1}$ ...
2
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1answer
51 views

Weak derivative of one parameter group and the domain of its generator

Let $U(t)=\exp(i t A)$ be a one parameter group generated by self-adjoint (unbounded) operator A. It is well-known that if $$ \lim_{t\rightarrow 0} \frac{U(t)\psi-\psi}{t} $$ exists then $\psi$ ...
0
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2answers
100 views

bounded linear functional on $\ell^{1}$, and its relation to $\ell^{\infty}$

Prove that a bounded linear functional $F$ on $\ell^1$ has representation $F(x)=\sum_{n=1}^{\infty}(c_{n}x_{n})$ where $c_{n} \in \ell^{\infty}$, and that $\|F\|_{*} = \|c_{n}\|_{\infty}$.
0
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1answer
42 views

Linear Operator: Boundedness

I'm stuck at: $\sup_{\overline{B_1}}\lVert T x\rVert\leq\sup_{B_1}\lVert T x\rVert$? For sure it holds: $\sup_{B_1}\lVert T x\rVert=\sup_{B_1\setminus\{0\}}\lVert x\rVert\lVert T \frac{x}{\lVert ...