# Questions tagged [operator-theory]

Spectrum, resolvent, numerical range, functional calculus, operator semigroups. Special classes of operators: compact, Fredholm, dissipative, differential, integral, pseudodifferential, etc.

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### Existence of a solution to an infinite dimensional Stratonovich SDE

Let $U,H$ be separable $\mathbb R$-Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,(\mathcal F_t)_{t\ge 0},\operatorname P)$ ...
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### Unital $C^{*}$ algebras whose all elements have path connected spectrum

A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$. What is an example of a non commutative ...
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### How much does the absolute value of an operator behave like an absolute value?

Recall that the absolute value of a bounded operator $T$ on a Hilbert space $H$ is the unique positive operator $|T|$ such that $$\||T|x\|=\|Tx\|$$ for all $x\in H$. It can be defined using the ...
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### Homotopy groups of Fredholm operators

If $X$ is separable complex Hilbert space and $\mathcal{F}$ the topological space of Fredholm operators on $X$, then it is well-known, that $$\pi_0(\mathcal{F}) = \mathbb{Z}\, ,$$ i.e. the connected ...
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### Is the set of separable quantum states closed?

Let $\mathcal H,\mathcal H'$ be Hilbert spaces (not necessarily separable). A "separable state" is a trace-class operator of the form $\sum_i \rho_i\otimes\rho_i'$ where $\rho_i,\rho_i'$ are positive ...
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### Non-empty resolvent set, then operator closed?

On Hilbert spaces, the following is true: Let $T$ be a densely-defined linear operator with non-empty resolvent set, then $T$ is closed. The obvious proof I see to show this uses explicitly the ...
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### Comparison of the absolute value of an operator with its positive parts

It is well known that the absolute value on operators does not satisfy the triangle inequality. My question is whether for all positive operators $P,Q \in B(\mathcal H)$ is there a universal ...
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### Strongly convergent operator sequence

Let $T_j$ be a sequence of compact operators on a Hilbert space $H$ which converges strongly to the identity, i.e., for each $v\in H$ the sequence $$\parallel T_jv-v\parallel$$ tends to zero. Is it ...
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Let $T$ be an (unbounded) self-adjoint operator. Assume that there is a bounded operator $S$ such that $TS=ST.$ For which kind of $f$ do we have that $f(T)S=Sf(T)?$ My thought was that using a ...
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Suppose $H_1$ and $H_2$ are two Hilbert spaces with dimension $n$ and $m$, for $x \in H_1 \otimes H_2$ consider $$\|x\|_\pi = \inf \left\{ \sum_{i=1}^n \|a_i\| \|b_i\| : x = \sum_{i} a_i \otimes b_i ... 2answers 2k views ### The letters of the word "ART" Edit: According to the Gelfand duality between topological spaces and commutative C^{*}algebras, I add some new tags. So the question is that what is the structure of  Ext (A,A) where A is ... 2answers 522 views ### Is \mathcal{B}^{\mathbb{Z}}(l^\infty(\mathbb{Z})) a commutative algebra? Consider l^\infty(\mathbb{Z}) the Banach space of bounded complex valued functions on the abelian group \mathbb{Z} with the supremum norm. It has a natural action by \mathbb{Z} given by (zf)(g):... 3answers 403 views ### Does the generator of a 1-parameter group of Banach space isometries know which elements are entire? Let X be a complex Banach space. Let (\sigma_t)_{t \in \mathbb{R}} be a 1-parameter group of linear isometries of X which is strongly continuous i.e. t \mapsto \sigma_t(x) is continuous for ... 2answers 631 views ### \zeta-function regularized determinants In (mathematical) physics in order to compute path integrals one often makes an infinite dimensional change of variables and uses infinite Jacobian as a purely formal expression. This step is done in ... 2answers 393 views ### If the cardinality of B(X), the space of operators on X, is continuum, must X be separable? Does there exits any non-separable Banach space X such that the size (cardinal number) of B(X), bdd linear operators on X, is just of the continuum? 2answers 335 views ### Differentiation of functions over graphs In short: There are various ways to define differentiation over a graph. I am trying to get the big picture, like a more complete and structured bestiary. Definitions. Let G=(V,E) be a directed ... 2answers 428 views ### Hölder continuity for operators Let x,y be positive real numbers then$$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$we obtain 1/... 1answer 404 views ### Commuting with an unbounded operator Let H be a Hilbert space. Let A be a closed unbounded operator, and let B\in B(H) be a bounded operator. Definition: A and B strong-commute if the partial isometry in the polar ... 2answers 904 views ### Can one hear the shape of a drum for operators? M. Kac in his famous paper "Can one hear the shape of a drum?" asked whether one can "hear" the area of the ambient domain by looking at the spectral picture. Although he was not the first who came up ... 1answer 861 views ### Existence of a projection operator onto subspace of Hilbert space Let V \subset H be Hilbert spaces with a continuous, compact and dense imbedding. Let \{w_j\}_j \subset V be a basis of V and of H (so finite linear combinitions are dense) which is not ... 1answer 268 views ### Feynman-Kac formula for lattice heat equation with non-diagonal potential Suppose that X is the continuous-time simple symmetric random walk on the lattice \mathbb Z^d (i.e., a simple symmetric random walk with i.i.d. exponential jump times), and let$$u(t,x):=\mathbf E\...
Let $H$ be $\ell^2({\mathbb N})$ and let $S:H\to H$ be the unilateral forward shift, so that $S^*S=I\neq SS^*$. Then a bounded operator $T:H\to H$ is Hankel if and only if it satisfies $TS=S^*T$. Let ...