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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Why this operator is not fredholm?

Define $f:S^{2}\to \mathbb{R}$ by $f(x,y,z)=z$. Let $D:=D_{\nabla f}$. As I learned from the following post this operator is not counted as a fredholm operator.( I did not underestand, ...
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35 views

Approximate unit of a separable C*-algebra

The following is a corollary of Takesaki's Operator Theory: My question: I do not know why the author says"there exists an n such that $||x(1-v_n)^\frac{1}{2}||<\epsilon$" . Please help me to ...
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1answer
35 views

An routine exercise about matrix norm

If $T_{n}\in M_{k(n)}(\mathbb{C})$ and $||T_{n}^{*}T_{n}-1_{k(n)}||\rightarrow0$, then $||T_{n}T_{n}^{*}-1_{k(n)}||\rightarrow 0$ too? (Here, $M_{k(n)}(\mathbb{C})$ denotes the $k(n) \times k(n)$ ...
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38 views

Norm of an element in a C*-algebra

The following is a part of a proof in Takasaki's Operator theory: Let $\epsilon>0$ and $A$ is a C*-algebra. For an $x\in A$, put $h=x^*x$ and $u_\epsilon=(h+\epsilon)^{-1}h$. We have then ...
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56 views

What is the image of operator exponential?

Given a Banach space $V$ and a bounded linear operator $A:V\to V$, the operator $e^A$ is bounded and invertible. When $V$ is finite dimensional, every invertible operator is of the form $e^B$ (one can ...
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33 views

Show that $H=\oplus_{\alpha \in \sigma(T)}\text{ker}(\alpha I -T)$

Suppose $T$ is an operator on a Hilbert space $H$ such that $\sigma(T)=\sigma_{p}(T)$ (point spectrum of $T$), and for each $\alpha \in \sigma(T)$, the corresponding eigenspace ker$(\alpha I-T)$ is a ...
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34 views

Tensor products, help with proof

Let $X$, $Y$ and $Z$ be Banach spaces. Let the space $X\otimes_{\epsilon}Y\otimes_{\epsilon}Z^{*}$ be the injective tensor product. The injective norm is defined as follows: ...
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16 views

Notation for operator that returns square of a function?

Let $F$ denote the vector space of all real-valued continuous functions on the real line. Suppose I have an operator $T:F\to F$ such that for any input function $f \in F$, $T$ returns the square ...
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41 views

The positive element in a C*-algebra

The following is a theorem of Conway's Functional Analysis: for the proof ($c\to a$), I think we can say: for $\lambda\in \sigma(a)\subset \Bbb R$, there is a character $h:C(\sigma(a))\to\Bbb C$ ...
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1answer
20 views

A question about Invariant subspaces of an algebra.

I feel that this is a very simple problem, but somehow I don't see the solution. I want to show that if $A$ is a subalgebra of $B(H)$ containing $1$ then if $B\in SOTcl(A)$, for every n, ...
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33 views

norm of a matrix that its entries are operators in B(H).

Let S is a subset of B(H). Define $M_2(S)=\{T= \left( \begin{array}{ccc} A & B \\ C & D \\ \end{array} \right) : A,B,C,D \in S\}$. what is the relationship between $||T||$ and ...
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39 views

Norm of Hardy-Littlewood maximal operator

We define Hardy-Littlewood maximal operator $M$ by \begin{equation} Mf(x)=\sup_{r>0} \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y)| dy \end{equation} where $B(x,r)$ denotes the ball centered at $x \in ...
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2answers
46 views

Self-adjoint elements in a C*-algebra

I have a simple question which confused me. Suppose $A$ is a C*-algebra. every $x\in A$ has a representation such as $x=a+ib$ where $a,b$ are self-adjoint elements of $A$. Also we claim that $x^*x$ ...
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39 views

*-isomorphism of a C*-algebra into an involutive Banach algebra is norm increasing

The following is a proposition of Takesaki's Operator theory: My question: How does he assume, considering the C*-subalgebra generated by k instead of $*$-Banach algebra B? Are we sure that the ...
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1answer
64 views

Show that a unitary operator is of the form exp(iA)

This is an exercise from chapter 2 in Conway's "A Course in Operator Theory": Show that every unitary operator on a Hilbert space can be written as $U=\exp(iA)$ for some Hermitian $A$. I tried to ...
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9 views

Monotone operator without symmetry?

A function $f : \mathbb{R}^{n} \rightarrow \mathbb{R}^n$ is monotone with respect to $P = P^\top\succcurlyeq 0$ if $$ \left( f(x) - f(y) \right)^\top P (x-y) \geq 0 $$ for all $x,y$. Now suppose that ...
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24 views

Compact Operator Inversion

Let I be a positive compact in $\mathscr{B}(\mathscr{H})$ (where $\mathscr{H}$ is some Hilbert space) then $I$ can be written (uniquely) as $A^2=I$ for some $A \in \mathscr{B}(\mathscr{H})$. My ...
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1answer
47 views

Does the supremum is finite?

Let $B(A)$ be the space of all bounded functions on a given set $A$, define a metric as follows $$d(x,y)=\sup \{|x(t)-y(t)| : t \in A \}.$$ Show that the supremum exists ?
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15 views

Adjoint Operator and transposed linear application

I have some problems to well understand the notion of adjoint. 1) First to simplify, let's choose an operator $u:E \to F$ defined on the whole domain from E (banach space) to F (Banach space). It ...
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14 views

Limit of an element in a unital C*-algebra

Let $A$ be a unital C*-algebra. Show that an element $x$ of $A$ is self-adjoint if and only if $\lim_{t\to 0}\frac{1}{t}(||1+itx||-1)$=0. My attempt: Suppose $x=x^*$. By functional calculus of x, ...
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1answer
21 views

Adjoint of differential operator in two variables

I would like to find the adjoint of the operator $$L = x \frac{\partial^{2}}{\partial y^{2}} \frac{\partial }{\partial x}.$$ I know the adjoint is the operator $L^{*}$ such that $$(Lu,v) = (u, ...
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30 views

There is no continuous function h on unit circle such that u=exp ih when spectrum u is entire unit circle

Let $\Gamma$ be the unit circle. Let u be the unitary element in $C(\Gamma)$ defined by $u(\lambda)=\lambda$. Show that there is no continuous function h on unite circle such that u=exp ih. ...
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23 views

functional calculus on a set of normal elements is continuous

Let $K$ be a compact subset of $\Bbb C$. Let $A_K$ denote the set of all normal elements $x$ with $\sigma_A(x)\subset K$. If $f$ is a continuous function on $K$, then the functional calculus :$x\in ...
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32 views

Confusion with Riesz

I was reading this article here and it claims that the covariance operator from a Hilbert space to itself exists by the "Riesz representation theorem". I don't seem to see the link between the Riesz ...
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22 views

Find an operator $Z$ in $H^1(0, \infty)$ with $\langle u,Zv\rangle = \int \bar{u}v dx$

I'm working with operators associated to bilinear forms. What I need to find is a continous, linear operator $T$ defined on $H^1((0, \infty))$ [note that $H^1 = W^{1,2}$ is the Sobolev space] such ...
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1answer
33 views

polar decomposition of multiplicative operator on L^2 induced by identity function.

We know that every operator in B(H) has a polar decomposition. $T=VP$ that $P=|T|$ and V is a partial isometry with initial space closure of ImP and final space ImT. How can i obtain polar ...
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1answer
11 views

Calculating the form domain of an operator

I am reading the book "Mathematical Methods in Quantum Mechanics" by Gerald Teschl and just came across the concept of a form domain. It is defined for non-negative operators i.e $<\phi,A \phi> ...
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28 views

Commutant of a C*-subalgebra of B(H)

In operator theory, we can prove that the commutant of $B(H)$ is $\mathbb{C} I$, where $I$ is the identity map. But a book states that every $C*$-subalgebra of $B(H)$ that contains the compact ...
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50 views

Does $\|f(T^*T)T^*\|_\infty = \|f(T^*T)(T^*T)^{\frac{1}{2}}\|_\infty$?

If $T:X\to Y$ is a compact operator and $X,Y$ are some Hilbert spaces, can we say that $\|f(T^*T)T^*\|_\infty = \|f(T^*T)(T^*T)^{\frac{1}{2}}\|_\infty$, where $T^*$ is its adjoint and $f$ some ...
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1answer
33 views

Find the operator norm

Let $T : \ell^2 \to \ell^2$ (involving complex numbers) be defined by $$ Tx := (x_1, x_1+x_2, x_3, x_3+x_4, x_5, x_5+x_6, \ldots). $$ What is $\|T\|$? Essentially I've tried : To find $M \geq 0$ ...
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25 views

Derivative of a function which is treated as a variable

I have got a function $f=f(x)$. The derivative is $\partial_xf$. There are applications in which it is reasonable to treat $f$ as another variable in a larger context. In my application I now need an ...
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1answer
28 views

an exercise about the projections.

There is an exercise in operator theory that says: If P and Q are projections on H that $||P-Q||<1$ then dimension of ImP and ImQ are the same. i cant understand what is the relation between the ...
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1answer
50 views

Norm of the Linear (Integral) Operator on $C[0,1]$

There may be some similar questions being asked but I didn't see anything that was quite what I was looking for so I'll ask the question here. Let $X = (C[0,1],||.||_{\max})$ and $T:X \to X = ...
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14 views

Homeomorphism between locally compact space $\Omega$ and maximal ideals space of $C_0(\Omega)$

the following is a proposition: If $\Omega$ is locally compact and $\Sigma$ is the maximal ideal space of $C_0(\Omega)$, then the map $x\to \delta_x$ is a homeomorphism. To prove it, the author ...
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68 views

Is there any multiplicative linear functional on B(H)?

If A is a Banach algebra, we say that $\Phi: A \longrightarrow \mathbb{C}$ is a multiplicative linear functional if $\Phi$ is nontrivial, linear and $\Phi(xy)=\Phi(x)\Phi(y)$. It is easy to see that ...
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1answer
36 views

A Representation of $C(X)$ is a positive map.

I quote this excerpt from Conway: "A representation $\rho:C(X) \rightarrow \mathcal{B(\mathcal{H}})$ is a $\ast$-homomorphism with $\rho(1)=1$. Also, $\|\rho\|=1$. If $f\in C(X)_+$, then $f=g^2$ ...
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66 views

Closed unit ball of B(H) is not compact in strong operator topology of B(H).

In operator theory we prove that closed unit ball of B(H) is compact in weak operator topology and is closed in strong operator topology. But a book of operator theory states that closed unit ball of ...
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1answer
46 views

The map $T\longmapsto \|T\|$ is not continuous in the strong operator topology of $\mathscr B(H)$

In the context of Strong and Weak operator topologies on $\mathscr B(H)$ there is an statement that says: the map on $\mathscr B(H)$ that $T\longmapsto \|T\|$ is not continuous in the strong operator ...
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1answer
30 views

an exercise about the spectrum of an element in Banach algebra.

An exercise of Banach algebra, section of spectrum has wanted the proof of this statement: Let $A$ be a Banach algebra and $x\in A$. Show that for every open set $U$ in $\mathbb{C}$ that contains ...
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44 views

Spectral radius of a normal element in a Banach algebra

I know that if $A$ is a C*-algebra, then $||x||=||x||_{sp}$ for every normal element $x\in A$. I like to find an example of a normal element $x$ in a involutive Banach algebra with $||x||\neq ...
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46 views

Is a contractive algebraic homomorphism between unital $ C^{*} $-algebras a unital $ C^{*} $-algebraic homomorphism?

We know that a $ C^{*} $-algebraic homomorphism from a unital $ C^{*} $-algebra $ A $ to a unital $ C^{*} $-algebra $ B $ is a linear multiplicative mapping that preserves units and respects the $ * ...
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1answer
26 views

Why is this statement true for two equivalent projections in $B(H)$?

In a book of operator theory it is stated that two projections $P$ and $Q$ in a von Neumann algebra $A$ are equivalent if there exist $V$ in $A$ that $V^*V=P$ and $VV^*=Q$. After this definition, it ...
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1answer
63 views

Spectrum of an element of a non unital C*-algebra

I know that spectrum of an element $x$ of a unital C*-algebra is nonempty. I like to find an example of a non unital C*-algebra that has an element with empty spectrum, if it exists. Motivation I ...
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3answers
115 views

Spectral radius of an element in a C*-algebra

The following is an proposition of Takesaki's Operator Theory: For any element $x$ of a Banach algebra ${\cal A}$, we have $$||x||_{sp}=\lim_{n\to \infty}||x^n||^{\frac{1}{n}}$$ Proof: My ...
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34 views

What operation is being done for this set of values?

I have a table that looks like the following: A B C A | B A C B | A C A C | C A B Some operation is being done between an element in the ...
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1answer
56 views

Why is the weak operator closure of a commutative $\boldsymbol{C^*\!\!\!\!-}$algebra also commutative?

In a book on Operator Theory there is the following statement: If $\mathscr A$ is a commutative $C^*$-subalgebra of $\mathscr B(\mathcal H)$, where $\mathcal H$ is a Hilbert space, then the weak ...
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1answer
23 views

Continuity of an integral operator

I'm stuck with this exercise: Let $A \subset \mathbb{R}$ be a measurable set. For each $f \in L^1(\mathbb{R})$ and $y \in \mathbb{R}$, let: $T(f, y) = \int_{A}{f(x-y)\mathrm{d}x}$. I have to show ...
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64 views

Does the integral operator, whose kernel is the indicator of the rhombus, belong to the trace class?

In connection with this question: Does the integral operator on $L^2(\mathbb R)$, whose kernel is the indicator of the rhombus $\{|x|+|y|<1\}$, belong to the trace class?
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108 views

What happens if we rotate the kernel of an integral operator?

Given an integral operator $K$ on $L^2(\mathbb R)$ with kernel $k(x, y)$, consider the integral operator $L$ on $L^2(\mathbb R)$, whose kernel has the form $k(\alpha x+\beta y, \gamma x+\delta y)$, ...
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3answers
44 views

Does an operator of x commute with the differential operator with respect to x?

While solving a problem in Quantum Mechanics I got an expression $ \frac{d}{dx}V(x)-V(x)\frac{d}{dx} $. The first term is just the derivative of the potential but the second one seems a bit weird. Is ...