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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Continuous functional calculus question

I have defined a continuous functional calculus on the bounded self-adjoint linear operators for functions continuous on the spectrum, and are defined on an interval. How do I deal with wanting to ...
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270 views

Does there exist a self-adjoint operator whose spectrum consists wholly of prime numbers?

The zeros of the canonical Riemann zeta function have been compared to the prime numbers, and they have a number of special, definite connections. The infamous zeros have also been conjectured to be ...
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252 views

Two different definitions of ellipticity

This is a question originating in another mathematics forum, matematicamente.it (in Italian). In literature one encounters the word elliptic in (at least) two different definitions. In what follows ...
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93 views

Strongly Continuous Multiplication on Bounded Subsets

In Pedersen's Analysis Now on page 171 there is an inequality which is given as an indication that multiplication is (jointly) strongly continuous on the subset $B(0,n) \times B(H) \subset B(H)$ (we ...
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4answers
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Double sum - Miklos Schweitzer 2010

There is a question in the Miklos Schweitzer contest last year that keeps bugging me. Here it is: Is there any sequence $(a_n)$ of nonnegative numbers for which $\displaystyle\sum_{n \geq 1}a_n^2 ...
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115 views

Principal eigenvalue

How is the principal eigenvalue of elliptic differential operator defined? Is it just a spectral radius?
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Easy Proof Adjoint(Compact)=Compact

I am looking for an easy proof that the adjoint of a compact operator on a Hilbert space is again compact. This makes the big characterization theorem for compact operators (i.e. compact iff image of ...
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677 views

spectrum of right shift operator on $\ell^2(\mathbb{Z})$

Consider the right shift operator on $\ell^2(\mathbb{Z})$. Is there a way of calculating (well, showing what it is since I already know it's $z$ s.t $|z| = 1$) its spectrum without reference to it ...
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Operator functions

Can anyone give me an example/application where a selfadjoint bounded operator on a Hilbert space are put through a function? I know how to define it but am struggling to find an application area for ...
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Operator norm and tensor norms

I have a linear operator $A\in\mathcal{L}(X,Y)$ where $X$ and $Y$ are some Banach spaces (or Hilbert spaces would also do, if that simplifies the answer.). The operator norm of $A$ is given by $$ ...
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232 views

Where does the notation $\mathrm{Ad}(U)$ for $a\mapsto UaU^*$ come from?

I have often seen, in the context of operator theory and operator algebras, the notation $\mathrm{Ad}(U)a=UaU^*$, where $U$ is a unitary operator on a Hilbert space $H$ and $a$ is a bounded linear ...
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248 views

Using Hahn-Banach in proving result about operators and their adjoints on Banach Spaces

This point has been giving me a lot of trouble. I am skipping around in learning functional analysis, and I've directly gone to the study of bounded operators without studying topology and basic ...
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523 views

Equivalence between the Operator Norm in the Space of Linear Functionals and Norm on Hilbert Space

I'm a little surprised that I'm stuck on this point, but I have my bad days. I am trying to understand a statement made in the proof of Theorem VI.1 in p.184 of Reed and Simon Volume I Functional ...
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Need an interesting linear bounded operator

Can anyone give me an example of such an operator (preferably, but not necessary, self-adjoint) on a Hilbert space? Not the usual ones you would find in a textbook (multiplication, integral, shift, ...
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325 views

Topologies and Continuity in Operator Theory

I am studying Operator Theory right now, but I have not had much exposure to topology. I am trying to pick it up along the way, and I am wondering about a probably simple point: What is the ...
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149 views

Cramer's rule for infinite dimensional vectors

For the equation $Ax = b$ in the finite dimensional linear space one can apply Cramer's rule to find $x$ if operator $A$ is linear. If there is an equivalent or a similar method for an infinite ...
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714 views

How can one relate inverse of a differential operator to an integral operator?

Informally speaking, the integral operator can be regarded as the inverse of some differential operator. In some very special case, finding the inverse of the differential operator is equivalent to ...
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Questions about a PDE: $-u_{txx}+u_{t}=u_{x}$, $u(x,0)=u_0(x)$, $x,t\in{\bf R}$

Consider the BBM equation: $-u_{txx}+u_{t}=u_{x}$, $u(x,0)=u_0(x)$, $x,t\in{\bf R}$. One may rewrite this equation as following $u_t=((I-A)^{-1}\partial_x)u$ where $Au=u_{xx}$ if $(I-A)^{-1}$ ...
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Compactness of a bounded operator $T\colon c_0 \to \ell^1$

Pitt Theorem says that any bounded linear operator $T\colon \ell^r \to \ell^p$, $1 \leq p < r < \infty$, or $T\colon c_0 \to \ell^p$ is compact. I know how to prove this in case $\ell^r \to ...
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what is/are the spectrum of operators and their applications

this is an educational question. can someone please explain with some simple examples: (1) what is/are the spectrum of operator (2) where it is useful For providing examples of spectrum of ...
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Spectrum of a linear operator

Let $\ell^2 =\ell^2(\mathbb{Z})$. Choose $\theta \in ]0,1[$ and set: $$Tx=(\theta x_{n-1} +(1-\theta)x_{n+1})_{n\in \mathbb{z}}$$ for each $x=(x_n)_{n\in \mathbb{Z}}\in \ell^2$ (thus $T$ is a convex ...
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On the isometry between bounded linear operators and the dual of nuclear linear operators

Let $H$ be a separable Hilbert space. Let $(e_i)_i$ be an orthonormal basis. For any bounded linear map $T$ we write, whenever possible $$\operatorname{tr} T := \sum_{i}^{\infty} \langle T e_i, e_i ...
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Uniform mean ergodic theorem

I'm working in Einsiedler and Ward's book on Ergodic Theory and in Exercise 2.5.4 they want to prove the following $$\lim_{N - M \to \infty} \frac{1}{N - M} \sum_{n = M}^{N - 1} U_T^n f \to P_T f.$$ ...
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Linear operator categories

Let's consider linear operators on the set of complex-valued functions to the same set. I wonder to which categories such operators can be classified. All linear operators I encountered so far fall ...
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Scale Operator $Uf(x)=f(kx)$

I am looking for an operator $U$, that can do this to a function: $$Uf(x)=f(2x).$$ In particular I am happy if there is an $U$ for the general case: $Uf(x)=f(kx)$. Does such an operator exist for ...
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Finding all linear operators $L: C([0,1]) \to C([0,1])$ which satisfy 2 conditions

As above, I'm trying to find all linear operators $L: C([0,1]) \to C([0,1])$ which satisfy the following 2 conditions: I) $Lf \, \geq \, 0$ for all non-negative $f\in C([0,1])$. II) $Lf = f$ for ...
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1answer
824 views

Cross product of operators

How to show that: $ (-i\nabla-eA)\times(-i\nabla-eA) = (ie\nabla \times A) $ i and e are constants A is a vector field $\nabla$ = vector differential operator
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What is operator calculus?

I watched the excellent interview with Richard Feynman: http://www.youtube.com/watch?v=PsgBtOVzHKI In the interview Feynman mention that he at young age re-invented operator calculus. I have searched ...
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If $(I-T)^{-1}$ exists, can it always be written in a series representation?

If $X$ is a Banach space, and $T:X \to X$ is a bounded linear operator with norm < $1$, then $I-T$ has a bounded inverse defined by $(I-T)^{-1} = \sum_{n=0}^\infty T^n$. Thinking in terms of a ...
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1answer
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Identities with Div, Grad, Curl

In physics there are lots of identities like: $$\nabla \times (\nabla \times A) = \nabla (\nabla \cdot A) - (\nabla \cdot \nabla) A$$ I'm wondering if there is an algorithmic algebraic method to ...