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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$\sigma$ strong topology and *-operation continuity

I know that if H is an infinite Hilbert space then *-operation is not continuous with respect to the $\sigma$ strong topology. Now, I have a question. If H is finite dimension, is *- operation ...
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4answers
77 views

Help with a 'simple' sum of linear operators and their adjoints acting on an orthonormal basis

Given an orthonormal basis $\{u_1,\cdots, u_n\}$ of a vector space $V$ I am asked to show that $$ \sum_{k=1}^n \|T^*u_k\|^2= \sum_{k=1}^n \|Tu_k\|^2 $$ for all $T\in \mathcal{L}(V)$ where $T^*$ ...
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218 views

Dyadic approach to Marcinkiewicz interpolation for Lorentz Spaces

In Exercise 21 , in a note for professor Terrence Tao on his own blog http://terrytao.wordpress.com/2009/03/30/245c-notes-1-interpolation-of-lp-spaces/ Exercise 21 Suppose we are in the ...
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1answer
52 views

Question about strong and norm convergence.

Maybe the answer to this question is so trivial that I can't see it: Why the strong convergence of operators (on an hilbert space) does not imply the norm convergence? Many books make this example: ...
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153 views

A problem about Linear Operator

$X$ and $Y$ are Banach Spaces.$ T$ is a linear bounded operator from $X \to Y$. There exists a real number $c$ which is positive, such that for any $y$ belonging to $T(X)$, there exists a $x$ which ...
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1answer
56 views

Showing this operator is densely defined

This is an example in Rudin's Functional Analysis, in the chapter on Unbounded Operators. Consider the right shift operator $V$ on $l^2$. It is an isometry and closed, and $I-V$ is one-one and so $V$ ...
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83 views

How to prove $e^{\log(1+x)}= 1+x$ by series expansion?

as the title says, i want to prove $e^{\log(1+x)}= 1+x$, by substitute $\log(1+x) = \sum _{i=1} ^{\infty} \frac{(-1)^{i+1}x^i}{i}$ and $e^x=\sum _{i=0} ^{\infty} \frac{x^i}{i!} $. Can some one help ...
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130 views

Prove that if $\lambda$ is an isolated eigen-value of $T=T^*$, then $\ker(T-\lambda)=E_{\{\lambda\}}H$

Here we have a self-adjoint, densely-defined operator $T$ on a Hilbert space $H$, and $E_M$ is the usual spectral projector for any Borel set $M$, i.e., $E_M=\int_M\text{d}E_t$ (this means, by ...
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1answer
34 views

About convergence of $(T_nR_n)$ when $(T_n),(R_n) \subset B(X)$

Let $X$ be a Banach space and $(T_n),(R_n) \subset B(X)$. (a) Prove that if $(T_n)$ converges strongly and $(R_n)$ converges strongly or uniformly, then $(T_nR_n)$ converges strongly (b) Prove that ...
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93 views

A simple question in functional analysis

A classical result, in functional analysis, says that if $T\in B(X)$, the function: $\lambda \rightarrow (\lambda I-T)^{-1}$ is analytic on $\rho(T)$(which is the resolvent set). If I fix an element ...
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113 views

A question about weighted forward unilateral shift operators

We define $$ B(x_{1}, x_{2},...)=(0, \frac{x_{1}}{2}, \frac{x_{2}}{3},...,x_{n})\in l^{2}(N), $$ How could be shown that that $B$ is a quasinilpotent?
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1answer
104 views

every denting point and strongly exposed point is extreme point

If $X$ be a Banach space and $K$ is a subset of $X$, then I want to prove Every denting point of $K$ is extreme point Every strongly exposed point of $K$ is extreme point $K$ is the closed convex ...
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91 views

About measurability of operators

I'm triyng without success, to find some examples of functions that: $\bullet$Are WOT-measurable, but not SOT-measurable. $\bullet$Are SOT-measurable, but not $||\cdot||$-measurable. I give the ...
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54 views

A question about tensor product of algebras of compact operators. [duplicate]

Let $\cal{H}$ be a separable Hilbert space and $\cal{K(\cal{H})}$ the algebra of compact operators acting on $\cal{H}$. Then $$\cal{K(\cal{H})}\otimes\cal{K}(\cal{H})\cong\cal{K}(\cal{H}\otimes H).$$ ...
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79 views

show that linear functional is unbounded

Let $F:C^{1}[0,1]\to \mathbb{C}$ be equipped with the supremum norm $||.||_{\infty}$, $F(f)=f^{\prime}(1)$. I am trying to show that $F$ is unbounded. Here is my idea. I take a sequence ...
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1answer
41 views

Spectral characterization of induced operator norm

Consider $\mathbb{R}^n$ with the $l^1$ norm and the induced operator norm $\| \cdot \|$ on linear maps $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$. Can $\|T\|$ be characterized somehow by the spectrum ...
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147 views

Norm of an operator induced by $L^2$-kernel is bounded by $L^2$- norm of the kernel

I am currently studying Hilbert Schmidt operators on my own using the book "Functional Analysis" (Vol 1) by Reed and Simon. There it is stated that a function $K \in L^2(M \times M, d\mu \otimes ...
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1answer
105 views

Spectrum of operator in infinite dimensional hilbert space

We know that if a complex hilbert space $H$ is separable, then for every compact set $K$, there exists a bounded linear operator $T : H \to H$ s.t $\sigma (T) = K$. My question is if this still holds ...
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42 views

“Almost orthogonalizing” matrices using a signature matrix

Suppose $A$ and $B$ are two real symmetric $n \times n$ matrices (If simpler, consider $A$ and $B$ to be 0/1 matrices, say, adjacency matrices of d-regular graphs). Then $||AB||_{op} \leq ...
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1answer
448 views

Fourier transform of inverse of Laplace operator

On $L^2(\mathbb{R}^n)$ consider the operator $(-\Delta+z)^{-1}$, for $\Delta$ being the Laplacian and $z\in\mathbb{C}$, on $\mathcal{D}(\Delta^{-1})$. How can one show that for $\phi\in ...
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2answers
29 views

Hemitian operator inequality

I am trying to find two Hermitian operators $A$ and $B$ (whose representations are $2 \times 2$ complex matrices) for which neither $A \leq B$ nor $A \geq B $ holds. Note that $A \geq B$ iff ...
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Smoothing operator on manifolds

I am reading John Roe "Elliptic operators, topology and asymptotic methods". On page 79 there is the definition 5.20 of smoothing operators. The definition is the following: "A bounded operator on ...
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2k views

is bounded linear operator necessarily continuous?

Let $U, V$ be separable Banach spaces. Suppose we have a bounded, linear operator $C : U\to V$. Questions are the following *) Shall $C$ be continuous since $V$ is a Banach space? *) In general, ...
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65 views

Nilpotents of order 2 are dense in the strong operator topology

I need some help with this homework problem. (The Hilbert space in question is $\ell^2(\mathbb{N})$.) I let $T \in \mathcal{B}(\mathcal{H})$ and looked at a basic nbhd in the strong operator topology ...
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1answer
55 views

Prove the left multiplication $L_A$ operator in $\mathcal{B}(\ell_2)$ is continuous?

Can someone assist me in showing that this operator is continuous as a map in the weak operator topology? I tried to do this with nets, but got stuck trying to "move" the operation inside the inner ...
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142 views

Prove Fredholm's theorem

I'm trying to show the Fredholm alternative, is one of Fredholm's theorems. Let $T$ is a compact operator and $T:E \to E$. where $E$ is a Banach space. We consider the equation: $$u-Tu=f ...
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1answer
197 views

Finite dimensional $C^*$-algebras

Show that if a $C^*$-algebra $A$ is reflexive as a Banach space, then $A$ must be finite dimentional. I tried to solve it; but, I could not. please help me for this exercise. Thanks a lot!
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160 views

Spectral theorem and projection

This should be simple, but I'm stuck. Let $A$ be an unbounded self-adjoint operator on a Hilbert space $H$. The spectral theorem says that there is a decomposition of $H$ into a direct sum for which ...
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119 views

Example of a net in $\mathcal{B}(\ell_2)$ that converges in the weak operator topology but not in the strong operator topology?

I need to show that the Strong Operator topology is strictly stronger in $\ell_2$ (space of complex sequences that are square summable). I can show that convergence in the strong operator topology ...
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86 views

Stone's theorem for 1-parameter groups of unitary multipliers?

Let $A$ be a nonunital C*-algebra and let $M(A)$ denote its multiplier algebra. Let $(u_t)_{t \in \mathbb{R}}$ be a strictly continuous 1-parameter group of unitary multipliers. That is, $u_t x \to x$ ...
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841 views

finite dimensional range implies compact operator

Let $X,Y$ be normed spaces over $\mathbb C$. A linear map $T\colon X\to Y$ is compact if $T$ carries bounded sets into relatively compact sets (i.e sets with compact closure). Equivalently if $x_n\in ...
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1answer
257 views

Isolated point in spectrum

"Any isolated point in the spectrum of a self-adjoint operator must be an eigenvalue". Is there an easy way to see this? The spectral theorem tells us that any self-adjoint operator is unitarily ...
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1answer
71 views

$\lambda_k \to 0$ implies $T$ is compact?

I am doing an exercise which asks to show that if $\{\varphi_k\}$ is an orthonormal basis in a Hilbert space with $T$ a bounded operator satisfying $T\varphi_k = \lambda_k \varphi_k$, then $\lambda_k ...
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1answer
93 views

Let $\delta$ be a linear functional equipped with the sup-norm. Show that $\delta$ is bounded and compute its norm.

Let $\delta:C([0,1])\rightarrow\mathbb{R}$ be the linear functional at the origin: $\delta(f) = f(0)$. If $C([0,1])$ is equipped with the sup-norm $$\|f\|_{\infty} = \sup_{0\leq x\leq 1}|f(x)|.$$ Show ...
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Operator in the commutant with certain property

If $T$ is a bounded operator with nontrivial kernel (in my case it is actually finite dimensional kernel and the operator is quasinilpotent) acting on an infinite dimensional Banach space, can one ...
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2answers
69 views

Proving that an operator $K$ is bounded and $||K|| = \max_{0\leq x\leq 1}\bigg\{\int_0^1|k(x,y)|dy\bigg\}$

Define $K:C([0,1])\rightarrow C([0,1])$ by $$Kf(x) = \int_0^1 k(x,y)f(y)dy,$$ where $k:[0,1]\times [0,1]\rightarrow \mathbb{R}$ is continuous. Prove that $K$ is bounded and $$||K|| = \max_{0\leq ...
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70 views

Does this show the norm of this operator is zero?

We have $$T: C[-1,1]:\to \mathbb{R}$$ $$T(f)=\int_{-1}^1 x f(x) dx$$ The norm considered in $C[-1,1]$ is $$||f||=\max_{x\in[-1,1]} |f(x)|$$ So using $$||T||=\inf\{M:||Tf||\leq M||f||\}$$ in this ...
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1answer
67 views

Identity plus finite rank has index $0$

I'm supposed to prove the strong Fredholm alternative in the form $$\text{Ind}(1-K)=0$$ for any compact operator $K:H\to H$ where $H$ is a Hilbert space and $$\text{Ind}(T):=\text{dim Ker }T+\text{dim ...
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1answer
36 views

Index of orthogonal projections

I'm reading a paper 'The index of a Pair of Projections' by Avron, Seiler and Simon at the moment and have a question about a definition: What do they mean by $C$ viewed as a map from $\text{Ran ...
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84 views

Multiplication operator with a function non-vanishing on the cantor set

Let $M_f$ be the multiplication operator, which acts on bounded functions $g$ on the unit interval as $g\mapsto fg$, with $f:[0,1]\rightarrow \mathbb{C}$ such that $f$ is nonzero only on the Cantor ...
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158 views

Limit of nth power of operator norm

I am given a compact operator $A$ which lives in a Banach algebra and whose spectral radius obeys $\rho(A)=\lim_{n\rightarrow\infty}||A^n||^{\frac{1}{n}}<1$. Now I want to prove that this implies ...
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1answer
124 views

Calculate the norm of this operator

$C[0,1]=\{ f : [0,1]\to [0,1], f$ continuous$\}$ $||f||_\infty=\max_{t\in [0,1]} |f(t)|$ $T:C[0,1]\to C[0,1]$ defined by $$(Tf)(t)=\int_0^1e^{s+t}f(s)ds$$ Find $||T||$ The usual way to do this ...
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Piecewise Continuous Fredholm Kernel

Suppose I have a ``Fredholm equation of the second kind" with kernel $$ K(x, y) = 1[x \geq u]k(x, y) $$ where k(x, y) is continuous. Let $f_n(x)$ be a smooth continuous approximation of $1[x \geq ...
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104 views

Show that the operator is bounded in $L_p$

Consider the operator $C$, acting on functions $f$ on the unit circle $S^1 = \left\{ z \in \mathbb C \mid |z| = 1 \right\}$ by the rule $$ (Cf)(z) = \frac{1}{2\pi i} ...
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74 views

equality of two operators…

Please help me with the following problem( give some hints or references): Let $X$ be a Banach space and $B(X)$ be the algebra of bounded linear operators on $X$. Suppose that $A$ and $B$ are two ...
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Exchangability of inner product and integral in bochner spaces

For the linear operator $e \in \mathcal{L}(V,V^{*})$, and sufficiently small $\delta s \in V := L^2(0,T,L^2(D))$ and $p \in V$ we have; $$ E[\langle e^{*}p, \delta s \rangle_{V}] = \langle E[e^{*} p ] ...
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62 views

Composing Translations and Reflections

Since $f(-x)$ is a reflection of $f(x)$ in the $y$-axis and $f(x+a)$ is a shift of $f(x)$ by $-a$ units, so for the longest time I've assumed $f(-(x+a))$ is a shift of $-a$ and then reflected in the ...
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72 views

rank of $A \otimes B$

For two matrices $A$ and $B$, what would be the rank of $A\otimes B$ as a matrix? Seems to me that $rank(A\otimes B) = rank(A)\cdot rank(B)$. But I don't see an elegant proof...
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1answer
121 views

Why is the set of compact operators closed in the space of all bounded operators between Banach spaces?

Let $X$ and $Y$ be Banach space. $B(X,Y)$ is the vector space of all bounded linear maps from $X$ to $Y$. Also, $K(X,Y)$ is the set of all compact operators from $X$ to $Y$. Why is $K(X ,Y)$ ...
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1answer
196 views

Holomorphic functional Calculus in Dunford and Schwartz

I am currently studying the spectral theory for bounded operators as described in the book "Linear Operators" by Dunford and Schwartz because I would like to obtain a better understanding of the ...