# Tagged Questions

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.

120 views

### eigenvalue question for a Toeplitz operator

Let $\phi$ be a nonzero function in $L^\infty(T)$ where $T$ is the unit circle. Let $M_\phi$ be the multiplication operator and $T_\phi$ be the Toeplitz operator. Show $T_\phi$ and $M_\phi$ have no ...
159 views

52 views

### Topology of $(\mathcal{A},*)$ determined by $\mathcal{A}_{sa}$?

Let $(\mathcal{A},*)$ be a $*$-algebra, we have the following observation: Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two norms on $\mathcal{A}$ such that the involution is an isometry with respect to ...
288 views

19 views

### noncommutative algebra

If we let $B$ be a noncommutative Banach algebra with unit $e$, then, obviously, $xy\not=yx$, but are they related? I suspect that the spectral radii of $xy$ and $yx$ are the same, but I couldn't ...
18 views

### R.Douglas “Banach Algebra Technique Operator Theory” - Chapter 2 issue

Just before 2.37 Corollary (Spectral Mapping Theorem) Douglas says: If $\varphi (z)= \sum_{n=0}^\infty a_nz^n$ is an entire function with complex coefficients and $f$ is an element of the Banach ...
38 views

16 views

61 views

38 views

### If $Q$ is a trace class operator on $U$, then each bounded, linear operator from $U$ to $H$ is a Hilbert-Schmidt operator from $Q^{1/2}U$ to $H$

Let $U$ and $H$ be Hilbert spaces $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ with finite trace $U_0:=Q^{1/2}U$ $L$ be a bounded, linear operator from $U_0$ to $H$ I ...
I'm having a problem in implementing the following problem: I have a quantum state so defined: $\left| \Psi\right>=\int \mathcal{A}(\omega_1,\omega_2)\hat{a}^\dagger_H(\omega_1)\hat{a}^\dagger_V(\... 0answers 18 views ### Is the generated semigroup by an elliptic operator be the transition semigroup? I am considering the time homogeneous Ito diffusion: $$dX_t=b(X_t)dt+\sigma(X_t)dW_t,$$ where$b,\sigma$currently are only assumed to be global Lipschitz for the existence of solution$X_t$. The ... 0answers 19 views ### Given an operator$Q$between a Hilbert space$U$and$L^2(ℝ^d;ℝ^d)$, is it possible to make sense of$U∋u↦(Qu)(x)$for a fixed$x∈ℝ^d$? Let$U$be a Hilbert space$H:=L^2(\mathbb R^d;\mathbb R^d)$for some$d\in\left\{2,3\right\}Q$be a Hilbert-Schmidt operator from$U$to$H$. I want that$\tilde Q(x)$, where $$\tilde Q(x):=... 0answers 20 views ### Two definitions of the operator \exp(x) in L^2(\mathbb R) The operator x acts on a dense subspace of L^2(\mathbb R) and is not bounded. So if we define \exp(x) via the power series \sum_{n=0}^\infty \frac {x^n}{n!}, convergence will not follow in the ... 0answers 32 views ### Minimality of the Spatial C*-Norm Given C*-Algebras A,B and x \in A \otimes_* B show that: (a) If (\tau \otimes_* \rho)(x)=0 for all \tau \in A_+^* and \rho \in B_+^*, then x=0. PS: \tau \in A_+^* means that \tau is ... 0answers 32 views ### Calculate the trace of T_nL where L\in L(H), T\in L(H,L(H)) and T_n:=\langle T,e_n\rangle_H for some ONB (e_n)_n of a Hilbert space H Let^1 (H,\langle\;\cdot\;,\;\cdot\;\rangle) be a Hilbert space over \mathbb R (e_n)_{n\in\mathbb N} be an orthonormal basis of H T\in\mathfrak L\left(H,\mathfrak L\left(H\right)\right) ... 0answers 27 views ### inverse of operator I want to calculate the inverse of the operator T=-\frac{\partial^{2} }{\partial x^{2}}-\frac{\partial^{2}}{\partial y^{2}}-(x^{2}+y^{2})+2( x\frac{\partial }{\partial y}-y\frac{\partial }{\... 0answers 22 views ### The necessity of defining the stable equivalence in the construction of the Grothendieck group K_0 I am confused about the process of the construction of the Grothendieck group K_0 in Murphy's C^*-algebras and operator theory section 7.1. Let A be a *-algebra and P[A]=\bigcup_{n=1}^\infty\... 0answers 38 views ### Proving that the point spectrum of T is not empty This problem comes from Rudin's Functional Analysis, Chapter 10. It's the problem 15. Suppose X is a Banach space, T\in\mathcal{B}(X) is compact, and \|T^n\|\geq 1 for all n\geq 1. ... 0answers 22 views ### semifinite von Neumann algebra, spectral projection, trace Let \mathcal{M} be a semifinite von Neumann algebra and \tau be a semifinite faithful normal trace on it. Let T,P_1,P_2\in \mathcal{M}, where P_1,P_2 are projections with P_1\perp P_2. Then, ... 0answers 14 views ### Show (Au)(x)=v_1(x) \langle u , v_1 \rangle+ v_2 (x) \langle u, v_2 \rangle where (Au)(x)=\int^\pi _0 2u(y) cos \bigg (\frac{x-y}{2} \bigg) dy Let A be a linear operator defined on L^2([0, \pi]) by$$(Au)(x)=\int^\pi _0 2u(y) cos \bigg (\frac{x-y}{2} \bigg) dy$$Where$0 \leq x \leq \pi$I am trying to show that$(Au)(x)=v_1(x) \...
Let $A = \int_{0}^{\infty} \lambda dE(\lambda)$ be the spectral decomposition of a selft-adjoint operator $A$ on a Hilbert space $H$. Then the restriction operator $P_{\lambda}$ for $A$ is defined by \$...