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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On calculating spectral projections

Consider following operator from this paper; Let $h$ be any function in $L^1$ relative to the measure $g(w)dw$ and $K\in\mathbb{C}$ Consider the linear operator $B$ on $L^1$ defined by $$(Bh)(x) = ...
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Eigenvalues of a Self-Adjoint Operator

It is easy to see that eigenvectors corresponding to distinct eigenvalues of a self-adjoint operator $T$ are mutually orthogonal. However, from this, it is supposed to be easy to see that for a given ...
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Laplacian on $\mathbb{S}^2$ has a pure point spectrum

Consider an operator $T = -\Delta + V(\theta)$ where $V(\theta)$ is $C^{\infty}$ and $T : C^{\infty}(\mathbb{S}^2) \subset L^2(\mathbb{S}^2)\rightarrow C^{\infty}(\mathbb{S}^2).$ I was wondering why ...
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How to calculate Hill's discriminant?

I am currently reading this paper on Schrödinger operators see here. On page 6 and 7 they talk about Hill's discriminant and how this is connected with the spectral properties. They also show some ...
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Compact projection on Banach space has finite rank

Let $E$ be a Banach space. Show that every compact projection has finite rank. I have no idea where to start.
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Numerical range of closure of operator

Let $B$ be an unbounded densely defined and closable operator. If $\mathcal{N}(B)$ is the numerical range, what can be said about the numerical range of its closure $\overline{B}$?
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computation example of Maximal operator

how to compute the answer M(f(x))? can anyone show a detail step? thank you!!
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Fixed Point Theorem in finite dimensional Euclidean space

A fixed point theorem says that: "any continuous mapping of $\mathbb{R}^n$ into a bounded subset of $\mathbb{R}^n$ has a fixed point". So consider $f: \mathbb{R}^n \rightarrow X \subset ...
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Multiplication operator on $L^2$ and spectral theorem.

Let's consider the multiplication operator by the independent variable in $L^2(\mu)$, where $\mu$ is a borel regular measure on $\mathbb{C}$: $Mf(z)=zf(z)$. I want to show that if $\phi$ is a borel ...
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29 views

Spectrum of multiplication operator by the independent variable in $L^2$

If $\mu$ is a regular Borel measure on $\mathbb{C}$ with compact support $K$, define $N_\mu$ on $L^2(\mu)$ by $N_\mu f=zf$ (the multiplication by the indipendent variable). An exercise in "Conway" ...
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Simple proof that $\|p(A)\|\le \sup_{|z|\le 1}|p(z)|$ for polynomials $p$ and $\|A\| \le 1$.

Let $\mathcal{H}$ be a complex Hilbert space, and let $A$ be a bounded operator linear operator on $\mathcal{H}$ with $\|A\| \le 1$. It is known that $\|p(A)\|\le \sup_{|z|=1}|p(z)|$ for all complex ...
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Nondegenerate representation

By the definition, we say a representation $(\pi,H)$ is nondegenerate if $cl[\pi(A)H ]= H$. Below I have two theorem, the first from Conway's Functional analysis and the second from Takesaki's ...
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32 views

Spectral measure and commutativity.

I want to prove that if $A\in B(H)$ and $N\in B(H)$ is a normal operator, and $AE(\Delta)=E(\Delta)A$, where $E$ is the spectral measure given by $N$ and $\Delta$ is a Borel subset of $\sigma(N)$, ...
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32 views

Some doubts concerning spectral theory.

Probably I'm saying something wrong (that's why the conclusions are strange) so please correct me! There is the continuous functional calculus for a normal element $N$ of a C*-Algebra. This means ...
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1answer
25 views

If $N$ is normal, $W^\ast(N)=\{N\}''$

I want to prove that if $N$ is normal, then $W^\ast(N)=\{N\}''$, where $W^\ast(N)$ is the Von Neumann algebra generated by $N$ and $\{N\}''$ is the bicommutant of $N$. for the inclusion $W^\ast(N) ...
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1answer
42 views

Why is $\int_{\mathbb{R}^3} |p\rangle \langle p| d\lambda(p)=id$?

As I have written in the headline, I am curious how the relation $\int_{\mathbb{R}^3} |p \rangle \langle p| d\lambda(p)=id$ that physicists use, where $|p\rangle$ is the eigenfunction to the ...
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1answer
35 views

Cyclic representation on $L^2(\mu)$

Show that if $(X,\Omega,\mu)$ is a $\sigma-$ finite measure space and $H=L^2(\mu)$, then $\pi:L^\infty(\mu)\to B(H)$ defined by $\pi(\phi)=M_\phi$ is a cyclic representation and find all the cyclic ...
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The spectrum of a product of operators

Suppose $A,B\in\mathcal{B}(\mathcal{H})$, where $\mathcal{H}$ is an infinite dimensional Hilbert space. In general, we know that there is no relationship between $\sigma(AB)$ and $\sigma(A)$ and ...
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Showing $||L|| = \sup_{x,\, y\, \in H,\, x,\, y\, \neq 0} \frac{|\langle Ax,y\rangle|}{||x||\cdot ||y||}.$

Prove that, for a Hilbert space $H$ and a linear bounded operator $L:H \to H$ such that the domain of $L$ is $H$, $$||L|| = \sup_{x,\, y\, \in H}_{ x,\, y\, \neq 0} \frac{|\langle ...
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spectrum of a convolution operator is connected

Let $k \in L^1(\mathbb{R}; \mathbb{C})$ and define a linear operator $T$ acting on $L^2(\mathbb{R}; \mathbb{C})$ by $$ Tf(x) := [k \ast f] (x) \ldotp$$ Linearity is obvious, while the fact that $T$ is ...
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Norm of $A$ is zer0 when $\Bbb H$ is Complex Hiblert Space

If $\Bbb H$ is a $\Bbb C$-Hilbert space and $A\in \Bbb B(\Bbb H),$i.e. bounded linear operator on $\Bbb H$ such that $\langle Ah,h\rangle=0$ for all h in $\Bbb H$, then $A=0$ For the proof of this ...
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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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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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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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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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55 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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32 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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When $\int_\Omega f(x)K(x,y)dx$ is injective

Let $T:X\to Y$ be a compact integral operator and consider $g=Tf$, i.e. $$g(y) = \int_\Omega f(x)K(x,y)dx$$ where $\Omega=[a,b]$ is some subset of $\mathbb{R}$. Are there some particular conditions on ...
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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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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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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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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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1answer
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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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35 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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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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*-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
50 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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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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1answer
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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45 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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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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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
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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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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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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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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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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
31 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 ...