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# Questions tagged [operator-theory]

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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### SG pseudodifferential operators, Fredholm implies elliptic

How can i prove that an operator $T_\sigma$ with $\sigma\in S^{m_1,m_2}$ ; $(m_1,m_2)\in(-\infty,\infty)$ $T_\sigma: H^{s_1,s_2,p}\rightarrow H^{s_1-m_1,s_2-m_2,p}$ which is Fredholm is elliptic? I ...
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### What is the square-root of the nonnegative self-adjoint operator $L^2\to L^2,f\mapsto\int f$?

Let $(E,\mathcal E,\mu)$ be a probability space. Since $\mathbb R$ is naturally embedded into $L^2(\mu)$, we may consider the operator $$U:L^2(\mu)\to L^2(\mu)\;,\;\;\;f\mapsto\int f\:{\rm d}\mu.$$ ...
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### If $A^{1/2}$ is the square-root defined in terms of the spectral decomposition, are we able to show that $\mathcal D(A^{1/2})\supseteq\mathcal D(A)$?

Let $H$ be a $\mathbb R$-Hilbert space, $A$ be a densely-defined nonnegative self-adjoint linear operator on $H$, $(\pi_\lambda)_{\lambda\in\mathbb R}$ denote the spectral family on $H$ corresponding ...
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### What is an example of an antilinear operator

A linear operator $L$ is defined as $$L\Big(c_1f_1(x)+c_2f_2(x)\Big)=c_1Lf_1(x)+c_2Lf_2(x)$$ where $c_{1,2}$ are complex numbers. An example is $d/dx$. What could be an example of an antilinear ...
While reading a paper on the topic 'Numerical solutions for generalized Black-Scholes equation', It is given that their numerical scheme can be executed explicitly by solving a linear system $\mathbf ... 1answer 34 views ### Operator norm of the self adjoint part Let$A=\frac{1}{2}(T+T^\ast)$be the self adjoint part of a bounded operator$T$on a Hilbert space. Let$f$be a polynomial. When do we have$\|f(A)\|\leq \|f(T)\|$in the operator norm? The ... 0answers 31 views ### Conditions for a subalgebra to be weakly dense Let$H$be a complex, separable Hilbert space and$\mathcal{B}(H)$denote the algebra of bounded linear operators on$H$. Let$A \subset \mathcal{B}(H)$by a subalgebra. I'm looking for some ... 0answers 27 views ### Adjoint vs Self-adjoint operators represented by matrices I want to see the difference between just adjoint and self-adjoint (hermitian) operator represented by matrices. If I have a matrix $$A= \begin{pmatrix} 1 & i \\ i & 1 \\ \... 0answers 33 views ### Dilation of a contraction in the connection with numerical range. Let T\in\mathcal{B(\mathcal{H})} be a contraction and X\in M_n with \Vert X\Vert\leq 1 s.t. W(X)\subseteq \overline{W(T)} where W(T)=\{\langle Tx,x\rangle :\Vert x\Vert=1\} is the numerical ... 1answer 44 views ### Norm of Ratio of Operators Suppose a(L) and b(L) are series of negative powers of the lag operator, that is$$a(L) = \sum_{j=1}^\infty a_jL^{-j},\quad\text{and}\quad b(L) = \sum_{j=1}^\infty b_jL^{-j}.$$Also suppose that ... 1answer 21 views ### Intuition for absolute value of a bounded operator in the context of polar decomposition Let T be a bounded operator on Hilbert space. The functional calculus for bounded symmetric operators defines a positive symmetric |T|=\sqrt{T^\ast T}. Different operators can have the same ... 2answers 67 views ### When is \mathbb{B}(\mathbb{X}) compact? Let \mathbb{X} be a (real or complex) Banach Space and let \mathbb{B}(\mathbb{X}) be the space of all bounded linear operators of \mathbb{X}. Let \tau_u be the Uniform Topology on \mathbb{B}... 1answer 69 views ### Mathematical operations order when using an operator I am not very familiar with operators (as I do not study mathematics) and I have just started a Quantum Mechanics course in a university. However, I am not sure what should be the precise order of ... 1answer 26 views ### two equivalent projections in matrix algebra Let A be a unital C^*-algebra and a be a positive element in A such that \|a\|\leq 1,show that p =\left [ \begin{matrix} a & (a-a^2)^{\frac{1}{2}} \\ (a-a^2)^{\frac{1}{2}} &... 0answers 19 views ### What is the difference between compositional inverse and multiplicative inverse in Quantum mechanic? Really am mixed , I have read a definition of bounded linear operator as it defined below A bounded linear operator U: H \to H on a Hilbert space H is called a unitary operator if it satisfies ... 1answer 35 views ### Schatten class operators form Banach algebra? I am reading about Schatten p-class operators. Denote by S_p(H) the space of all bounded linear operators with finite Schatten p-norm. I know that S_p(H) is an ideal of B(H) and is a Banach ... 1answer 26 views ### comparision of two positive invertible elements in a C^*-algebra Suppose A is a unital C^*- algebra,a,b are any two positive invertible element in A,do there exist s,t>0 such that sa\leq b\leq ta? 1answer 502 views ### Does \sigma(T) = \{1\} and \|T\| = 1 imply that T is the identity? Suppose that T is a bounded linear operator on a complex Banach space X and that we know that \sigma(T) = \{1\} and \|T\| = 1 (i.e. the spectrum of the contraction T consists only of a single ... 0answers 30 views ### Is this (Hibert-Schmidt) integral kernel bounded? It is well known in the theory of bounded operators on L_2(\mathbb{R}, \mu) that the operator$$ Tf(x) = \int_\mathbb{R} k(x,y) f(y) \: dy$$is compact whenever$k(x,y) \in L_2(\mathbb{R}\times \...
I am trying to find the infinitesimal generator of the average absolute deviation of the Poisson process. i.e. the generator of $X_n(t)=\dfrac{Y(n^2t)-\lambda n^2t}{n}$. Observe that even though it'...