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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Spectral Measures: Numerical Range

Given a Hilbert space $\mathcal{H}$. Consider a normal operator $N:\mathcal{D}(N)\to\mathcal{H}$. The goal here is to prove: $$\langle\sigma(N)\rangle=\mathcal{W}(N)$$ By a previous result one has: ...
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Normal Operators: Numerical Range

Disclaimer: As I realized in the comments that this works for normal operators I decided to modify this question. Besides, I got the proof now - thanks to T.A.E.! Prove that for normal operators the ...
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Domain Issue: Notation

Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$A:\mathcal{D}A\subseteq\mathcal{H}\to\mathcal{K}$$ It is well known that:* $$A=A^{**}\iff ...
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Example of non-normal operator whose all eigenvalues are real

Does there exist a non normal operator whose all eigenvalues are real.
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How can I solve the following exercise [closed]

Prove that a linear operator $T:X\rightarrow Y$ is bounded if and only if it maps sequences that converge to zero to bounded sequences .
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Spectral Measures: Existence

Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ By the previous threads: $$Z=N\sqrt{(1+N^*N)^{-1}}\quad ...
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Normal Operators: Retransform

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ Construct the operator: $$Q:=(1+N^*N)^{-1}:\quad Z:=N\sqrt{Q}$$ By the previous ...
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Normal Operators: Transform

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ Construct the operator: $$Q:=(1+N^*N)^{-1}:\quad Z:=N\sqrt{Q}$$ Then it is ...
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How exactly does one define the “spectral measure” of an operator?

I am seeing kind of different definitions of "spectral measure" at different places and its not clear to me as to what is the universal idea. It would be great to get some "standard" definition. In ...
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Spectral Measures: Scale Operators

This thread is only Q&A. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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16 views

Spectral Measures: Scale Forms

This thread is only Q&A. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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Normal Operators: Examples

Given the Hilbert space $\mathbb{C}^2$. Consider bounded opertors: $$N:\mathbb{C}^2\to\mathbb{C}^2:\quad\|N\|<\infty$$ Then there are some with: $$N\neq N^*\quad N^*N=NN^*$$ What examples are ...
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Boundedness of $A$ in the operator equation $Au = f$ of $-\Delta u(x)=f(x)$.

We consider the boundary value problem on a bounded, open domain $\Omega \subset \mathbb R^n$ determining $u : \Omega \rightarrow \mathbb R$ such that $$-\Delta u(x)=f(x), \qquad ...
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Stability of ground state under positive (not relatively bounded) perturbations

This is about positive perturbations that are not necessarily relatively bounded, but where the perturbed operator is known (by some independent proof) to be self-adjoint. Is this a known result (or ...
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1answer
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Sobolev spaces, extensions and embeddings

I have the following statement whith an argumentation which I do not understand. Fix integers $k,l$ such that $0\leq l\leq k$. Then the identity map on $C^\infty(\mathbb{T}^d)$ extends to the ...
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1answer
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Reducing Spaces: Domain

Problem Given a Hilbert space $\mathcal{H}$. Consider normal operators: $$N:\mathcal{D}(N)\subseteq\mathcal{H}\to\mathcal{H}:\quad N^*N=NN^*$$ Denote for readability: ...
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51 views

Spectral Measures: Scale Spaces

This thread is only Q&A. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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112 views

Spectral Measures: Scale Embeddings

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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1answer
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Are lattice operations in set of orthogonal projections in Hilbert space continous?

Let $H$ be Hilbert space and denote set of all orthogonal projections in $H$ by $\Pi$. Then $\Pi$ can be given structure of a lattice. We partially order it by declaring $P \leq Q$ if $Q-P$ is ...
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If $(\lambda_n)_{n=1}^\infty$ is a bounded sequence, then there is a bounded linear operator $A$ on a Hilbert space $H$ such that $Ae_n=\lambda_n e_n$

If $(\lambda_n)_{n=1}^\infty$ is a bounded sequence, then there is a bounded linear operator $A$ on a Hilbert space $H$ such that $Ae_n=\lambda_n e_n$ for all $n\in \mathbb{N}$. Let ${e_n}$ be a ...
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Closed convex hull of unitaries

If a C*-algebra ${\cal U}$ contains a non-unitary isometry $S$, show that $$\|S-A\|>\frac{1}{2n}$$ for every $A=\sum_{i=1}^n \lambda_iU_i$ which is the convex combination of $n$ unitaries. Thanks ...
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Show that a subspace is closed in a Hilbert space $H$

If $T$ is a bounded linear operator in a Hilbert space $H$, and $T$ is self-adjoint and is equal to its inverse, how can I show that $\widehat{H} = \{h + Th : h \in H\}$ is closed? If I consider the ...
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Wave Operators: Summary

This thread is Q&A. Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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Normal operators on a hilbert space over the reals - does $norm(Tx)=norm(T^*x)$ imply $T$ normal?

The title states the question. It's easy to prove the result for scalars C via polarisation identities but I don't think the same method works in the real case: Let $S=TT^*-T^*T$ then one obtains ...
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About the trace class operators and their motivation

What is the motivation for trace class operators? Can anybody suggest the most general and standard reference that includes Schatten $p$ class operators as well? I have the following references. ...
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Polar Decomposition: Ranges

This is just a note. Given Hilbert spaces $\mathcal{H}$, $\mathcal{K}$. Consider a closed operator: $$A:\mathcal{D}(A)\to\mathcal{K}:\quad A=A^{**}$$ Construct its modulus: ...
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Prob. 9, Sec. 4.3 in Kreyszig's Functional Analysis Book: Proof of the Hahn Banach Theorem without Zorn's Lemma

Here's Theorem 4.3-2 (i.e. the Hahn Banach theorem for normed spaces): Let $f$ be a bounded linear functional defined on a subspace $Z$ of a normed space $X$. Then there exists a bounbed linear ...
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Prob. 6, Sec. 4.3 in Kreyszig's Functional Analysis Book: What are all possible extensions?

Here's Theorem 4.3-2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $X$ be a normed space, let $Z$ be a subspace of $X$, and let $f$ be a bounded linear functional ...
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1answer
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Wave Operators: Calculus

Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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Wave Operators: Unitarity

This thread is Q&A. Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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Normal operator and real eigenvalues [duplicate]

If all eigenvalues of normal operator are real, will it imply operator is self adjoint
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Intuitive understanding of the operator norm?

I understand various vector norms, but I don't understand operator norms. Specifically, norms on linear operators. Can anyone explain them?
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Wave Operators: Hamiltonian

Reference This is taken out of M. Reed and B. Simon, Scattering Theory. Problem Given a Hilbert space $\mathcal{H}$. Consider a free Hamiltonian $H_0$ and a perturbed one $H$. Introduce the wave ...
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Wave Operators: Reducibility

Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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1answer
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Wave Operators: Isometry

This thread is only Q&A. Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}H_\#\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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How to prove a set is a core for an infinitesimal generator

I am trying to prove that the set $D=\bigcap_{n\geq 1} \{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f(\frac{k}{\sqrt{n}})g_n (k)\}$ is a core for the infinitesimal generator of reflecting brownian ...
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Spectral Measures: Reducibility

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
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Wave Operators: Adjoint

Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}H_\#\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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1answer
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Is $B(H)$ the weak-$*$ closure of $K(H)$?

I am getting the following result: If $H$ is a Hilbert space, then the weak-$*$ closure of $K(H)$, the space of compact operators on $H$, is $B(H)$, the space of bounded operators on $H$. Is this ...
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Isometries: Weak vs. Strong

Given a Hilbert space $\mathcal{H}$. Consider isometries: $$R_\lambda\in\mathcal{B}(\mathcal{H}):\quad R_\lambda^*R_\lambda=1$$ Then it follows: $$R_\lambda\rightharpoonup R\implies R_\lambda\to R$$ ...
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35 views

Show by example that $AB=I$ does not imply that $BA=I$, with $I$ being the identity operator on $Y$. What is a suitable $Y$ for this to hold?

Let $A$ and $B$ be bounded linear operators on a normed space $Y$ into $Y$. Show by example that $AB=I$ does not imply that $BA=I$, with $I$ being the identity operator on $Y$ Here is what I have ...
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Hamiltonian: Derivative

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard the evolution: $$A=A^*:\quad A(t):=e^{-itH}Ae^{itH}$$ Suppose invariance: ...
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Hamiltonian: Weak Convergence

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a sequence: ...
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What is the relation between the matrix of a bounded linear operator and that of its adjoint?

Let $X$ and $Y$ be finite-dimensional normed spaces, both real or both complex, and let $\dim X = n$ and $\dim Y = m$. Let $E \colon= ( e_1, \ldots, e_n )$ be an ordered basis for $X$, and let $F ...
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On the square root function of matrices

Let $A, B$ be positive definite matrices and let $P$ be an orthogonal projection. If $A \leq PBP,$ does it follow that $$ A^{1/2} \leq PB^{1/2}P?$$
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Weyl sequence for closure of an operator

I'm trying to solve following exercise and need some hints. Let $A= \bar{ A_0 }$ be closure of $A_0$ - a densely defined operator. Suppose $f_n \in D(A)$ is Weyl sequence for $z \in \sigma (A)$. Show ...
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1answer
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cauchy's integral formula in operator theory

Can you prove cauchy's integral formula based on the assumptions in Conway book, operator theory, VII, 4.2? 4.2. Cauchy's Integral Formula If $\mathcal X$ is a Banach space, $G$ is an open subset ...
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Uniform closure of polynomials

What is the meaning of "uniform closure of polynomials"? I have seen it in Conway's Functional Analysis book VII § 5.
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Prob. 8, Sec. 4.5 in Kreyszig's functional analysis book: The inverse of the adjoint operator is the adjoint of the inverse operator

Let $X$ and $Y$ be normed spaces, both real or both complex, let $B(X,Y)$ denote the space of all the bounded linear operators $T \colon X \to Y$, and let $T^\times$ denote the adjoint operator of ...
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How to apply Theorem 4.3-3 in the proof of Theorem 4.5-2 in Kreyszig's functional analysis book?

Here's Theorem 4.3-3 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $X$ be a normed space and let $x_0 \neq 0$ be any element of $X$. Then there exists a bounded ...