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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An exercise of positive element in C*-algebra

Let $A$ be a unital C*-algebra and $\{b_{n}\}$ be a positive invertible sequence in $A$. If $||1_{A}-b_{n}||\rightarrow 0$, can we conclude $||1_{A}-b_{n}^{-\frac{1}{2}}||\rightarrow 0$ ?
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Lack of a polar decomposition

Prove that the left and right shifts on $l_{2}$ have no polar decomposition (i.e. $UP$ where $U$ is unitary and $P$ is positive).
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51 views

Operator Norm = 1

Let there be a linear map $T$ such that $T: \mathbb R^n\to\mathbb R^m$. The operator norm $\lVert \cdot\lVert_{op}$ of $T$ is then defined as the largest value of $c$ for which $\lVert T(\vec v)\lVert ...
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116 views

square root of positive operators

It $T, S$ are positive operators, do we have that $\sqrt{TS}=\sqrt{T}\sqrt{S}$? Are there any basic rules that hold for square roots of positive numbers that don't hold for positive operators?
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A problem of weak* continuity in relation with semigroups

Let $(\Omega,\Sigma,\mu)$ be a probability space. Let $\mathcal{A}$ ba a $\sigma$-subalgebra of $\Sigma$. We denote by $\mathbb{E} \colon L^\infty(\Sigma) \to L^\infty(\mathcal{A})$ the associated ...
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The spectrum of the operators

Let $X, Y$ be the Banach space, and $T_{1}: X\rightarrow X$ and $T_{2}: Y\rightarrow Y$ be the bounded linear operators. Then what is the relationship between $\sigma(T_{1})$, $\sigma(T_{2})$ and ...
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functional calculus and spectral measure

Let $T$ be a normal operator and $f$ be a bounded borel function on ${\sigma}(T)$. If $E_{T}$ and $E_{f(T)}$ are the spectral decompositions of $T$ and $f(T)$ respectively, prove that for any borel ...
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Operator's comparison

I have found the following two concepts: $\bullet$ Let $L$ be a linear operator in a Hilbert space $H$. The operator $B$ is said to be $L$-compact if $D(L)\subset D(B)$ and for any ...
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232 views

Exponential of a function times derivative

Exponential of a derivative $e^{a\partial}$ is simply a shift operator, i.e. \begin{equation} e^{a\partial}f(x)=f(a+x) \end{equation} This can be easily verified from a Taylor series \begin{equation} ...
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81 views

generalized eigenspace direct sum

Similar to the way an infinite dimensional hilbert space can be written as a direct sum of eigenspaces of a normal compact operator, I was wondering whether it can be written as a direct sum of ...
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36 views

The norm of operator matrix

Let $H$ be a Hilbert space and $B(H)$ be the bounded linear operator on $H$, for $T\in B(H)$, if $T=\left(\begin{array}{ccc} 0 & B \\ A & 0 \\ \end{array}\right)$ on $H=M\oplus ...
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35 views

spectral measure and integral query

I have proved the 'resolution of the identity' for a normal operator, namely that there is a unique spectral measure E such that $\int_{{\sigma}(T)} {\lambda}\,dE=T$ If (${\lambda}_{n}$) is the ...
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Borel functional calculus

For a normal operator T, we have a resolution of the identity $\int_{{\sigma}(T)} {\lambda}\,dE=T$. If $T$ is in addition compact , we have that $\sum_{n=1}^{{\infty}}{\lambda}_{n}\langle ...
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43 views

The operator matrix on Hilbert space

Let $H$ be a Hilbert space and $P$ be the projection operator, then $H= P(H)\oplus (1-P)(H)$. Hence, for each $T\in B(H)$, we have $$T=\left(\begin{array}{ccc} PTP & PT(1-P) \\ (1-P)TP ...
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Core for an unbounded operator.

A symmetric operator $T$ is called essentially self-adjoint if its closure $T$ is self-adjoint. If $T$ is closed, a subset $D \subset D(T)$ is called a core for $T$ if $\overline {T\upharpoonleft D} ...
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A easy question on projection operator

Let $H$ be a Hilbert space and $B(H)$ be all the bounded linear operators on $H$, for arbitrary $T\in B(H)$, if $\{P_{i}\}$ is an increasing net of finite-rank projection, can we conclude $P_{i}TP_{i} ...
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Projection operator in Hilbert space

Let $H$ be a Hilbert space, can we find an increasing net of finite rank projections which converge to the identity in the strong operator topology? And I think if $H$ is separable, we can find an ...
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Prove that — the range $R(T)$ of a bounded linear operator $T:X\to Y$ need not be closed in $Y$

Prove that the range $R(T)$ of a bounded linear operator $T:X\to Y$ need not be closed in $Y$.
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Rank of sum of projections

Let $(\varphi_j)$ be a linear independent sequence of elements of a Hilbert space, not necessarily orthogonal, but such that $$Kf := \sum_{j=1}^\infty \langle\varphi_j, f\rangle\varphi_j$$ converges ...
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83 views

Spectral decomposition of normal operator

Define $T$ from $L_{2}(R)$ into itself by $T(f)(t)=f(t+1)$. Show that $T$ is normal and finds its spectral decomposition. I've shown that $f$ is normal (in fact it's unitary) but how do I find its ...
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54 views

formula for the norm of a normal operator

In Rudin's Functional analysis, he does a theorem which shows that for a normal operator $\Vert T\Vert=\sup\left\{|\langle Tx,x\rangle|\colon \Vert x \Vert \leq 1\right\}$. Why can't $\Vert x \Vert ...
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27 views

Boundedness of a closed operator

Can I get any help with this problem: Let $X, Y$ be Banach spaces, let $D$ be a subspace of $X$, and let $A \colon D \to Y$ be a closed linear operator. If $D$ is a closed subspace of $X$, ...
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Showing that the space of Hilbert-Schmidt operators form a banach space.

How do i show that the set of Hilbert-Schmidt operators $HS(H) = \{T \in B(H) \; : \; \sum^{\infty}_{n=1}\|Te_n\|^2 < \infty \}$ for some countable ONB $\{e_n\}$, on a separable Hilbert Space H, ...
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I want to show that some subset of $C([0,1])$ is equicontinous

First why the problem appeard. I want to show that the linear and continuous operator $T:C([0,1])\rightarrow C([0,1])$ , $ (Tf)(t)=\int_{[0,1]}k(t,s)f(s)ds$ where $k:[0,1]^2\rightarrow\mathbb R$ is ...
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Fredholm operators and their applications

What are the applications (and possibly generalizations) of fredholm operators in partial differential equations?
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Polar decomposition corollary

Let $T$ be a compact operator on an infinite dimensional Hilbert space. Let $|T|=(T^*T)^{0.5}$. By the polar decomposition theorem there is a partial isometry $S$ of the closure of Im$(|T|)$ such that ...
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Relation between $K$-theories

I apologize in advance if this question is too vague/general. I am curious to know how all of the different $K$-theories are related (algebraic $K$-theory, topological $K$-theory, twisted $K$-theory, ...
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Spectral decomposition of $TT^*$

On $l_{2}$ let $T$ be given by $Te_{n}=\frac{e_{n+1}}{n+1}$ where $(e_{n})_{n\ge1}$ is the canonical orthonormal basis. Find the spectral decomposition of $TT^*$. I find that ...
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Completeness and separability of $B(X)$ (bounded linear operators on $X$)

Assume $X$ is a separable Banach space with norm $|| \cdot||$. Consider $\{f_n\}_{n \in N}$ a countable dense subset of $X$ and equip $B(X)$ (bounded linear operators on $X$) with the following ...
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Does this sequence of operators converge in norm or strongly?

Let $H$ be a Hilbert space and $\mathcal{L}(H)$ the set of all bounded linear operators $L:H\to H$, equiped with the usual norm $\|\cdot\|_{\mathcal{L}}$. Let $T:D(T)\subset H\to H$ be a ...
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Baker-Hausdorff Lemma from Sakurai's book

I'd like to show that, given to hermitian operators $A,G$ on a Hilbert space $\mathscr{H}$, the following identity holds: $$ e^{iG\lambda}A e^{-iG\lambda} = A + i\lambda [G,A] + ...
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Has this theorem (on existence of inverse) an analogous for unbounded operators?

Let $S,T:X\to X$ be bounded linear operators, where $X$ is a Banach space. It's a consequence of the Banach Fixed Point Theorem that if $T$ is invertible and $\|T-S\|\|T^{-1}\|<1$ than $S$ is ...
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Image and Kernel of a certain bounded operator

Consider and Hilbert Space $X$, $T\in B(X)$ and a scalar $\mu$ s.t. $|\mu|=||T||$ By a simple argument I deduced that $\ker(\mu I- T)=\ker(\bar\mu I-T^*$) where $*$ denotes the adjoint. I am then ...
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Largest/smallest cross-norm: A simple question about cross-norms on tensor products of Banach spaces.

This is a very simple dumb question as I’m completely new to the topic. I was reading Wikipedia’s entry on “Topological Tensor Products”, and there’s one thing I’m confused about. Let $ A $ and $ B $ ...
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In which cases the spectrum of an operator contains only eigenvalues?

Let $X\neq \{0\}$ be a complex normed spaces (not necessarily finite-dimensional) and $T:D(T)\subset X\to X$ a linear operator (not necessarily bounded). I would like to know under what conditions can ...
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Conditions on integral operator to be in $L^{2}$

Suppose we have an open set $\Omega$ in $\mathbb{R}^{n}$ and for every $x\in\Omega$ a function $T\left(x,\cdot\right)\in L^{2}\left(\Omega\right)$. If for $f\in L^{2}\left(\Omega \right)$ we consider ...
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Eigenvalues and eigenvectors of a nonlinear operator

I have found a few nice answers to the question: "Why are eigenvalues and eigenvectors useful." I can imagine that knowledge of eigenvectors (-values) for a general nonlinear operator is worthless. ...
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Application of the spectrum of an operator

http://en.wikipedia.org/wiki/Spectrum_of_an_operator What is the application of the spectrum of an operator
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Show the Volterra Operator is compact using only the definition of compact

The Volterra operator $V:L^{2}[0,1]\rightarrow L^{2}[0,1]$ is defined by $(Vf)(x)=\int_0^xf(t)dt$. I am wondering if it can be shown that $V$ is compact by definition - that is, either that $V$ ...
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Spectral Mapping Theorem

Spectral mapping theorem is as follows: https://math.uc.edu/~halpern/Matrix.methods/Homatrixmethods/Spectralmappingthm.pdf Is Spectral mapping theorem true for point spectrum ?
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Does these two operators commute?

It is an exercise, $z\in\mathbb T$, which is the unit circle $$m(z)=\sum_{k\in\mathbb Z}a_kz^k$$ set $$Sf(z)=\frac{1}{\sqrt N} m(z)f(z^N)$$ So the adjoint operator is ...
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Positive compact operator has unique square root.

Let H be a hilbert space and T be a compact positive operator so that by the spectral decomposition theorem, $T=\sum_{n=1}^{\infty}{\lambda}_{n}\langle x,e_{n}\rangle e_{n}$ where the $e_{n}$ are the ...
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Unital free semigroup and operators

Let $F_n^+$ be a unital free semigroup generated by $1,...,n$. Let $\alpha= i_1...i_k$ where $i_1,...,i_k\in \{1,...,n\}$ and put $T_{\alpha}:=T_{i_1}...T_{i_k}$ where for any $ i_j\in \{1,...,n\},~ ...
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Spectral theory and sequences: is this fact a general truth or does it depend on the operator?

Let $\lambda\in\mathbb{R}\setminus\{0\}$, $\textbf{i}$ the imaginary unit, $H$ a Hilbert space, $T:D(T)\subset H\to H$ a invertible densely defined linear operator such that $T^{-1}$ is bounded, ...
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Show these operators converge to a particular limit

Let $H$ be a Hilbert space, and $T$ be a operator on $H$ of the form $T=\sum_{n=1}^{\infty}{\lambda}_{n}<x,e_{n}>e_{n}$ where $e_{n}$ are the eigenvectors of $T$ and an orthonormal basis of H ...
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Comparison of operators

I have found the following two concepts: $\bullet$ Let $L$ be a linear operator in a Hilbert space $H$. The operator $B$ is said to be $L$-compact if $D(L)\subset D(B)$ and for any ...
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1answer
37 views

An exercise in operator theory

Let $H$ be a Hilbert space and $P$ be a projection to a finite dimensional subspace $K$ of $H$, for a $T\in B(H)$, if $||PTP||=1$, then, for arbitrary $\epsilon>0$, there exists a vector $\alpha$ ...
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55 views

Invariant convex subset

Let X be a banach space and $T$ a bounded operator on X with $||T||$ less than or equal to 1. If $T$ is an isometry and $r(T)$<1 show that there is a closed convex non-zero proper subset $C$ of the ...
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Powers of a closed range operator

suppose that $S$ and $S^2$ are operators with closed range. Does it follow that $S^n$ is an operator with closed range for all natural numbers $n$?
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Kernel closed implies continuous operator?

Is closed kernel sufficient for linear operators to be continuous? Counterexample? Thx, Alex