# Tagged Questions

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

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### tr(ab)=tr(ba), part 2.

This is a Banach space version of Andre Henriques' question Trace Question for Hilbert spaces. Let $a:X\to Y$ and $b:Y\to X$ be bounded linear operators between Banach spaces s.t. $ba$ and $ab$ ...
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### A problem in functional calculus

This is embarrassing, I think it must work, but I can't see how to prove it works. If anyone knows enough functional calculus of operators on a Hilbert space to tell me how to do it, I would be very ...
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### An equivalence relation on the space of polynomials in one complex variable

Let $P(z)$ be a polynomial with complex variable $z$. We consider the following distribution for the roots of $P(z)=0$: the distribution is a triple $(n_{1},n_{2},n_{3})$ where these integers are ...
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### Is there an operator algebraic reformulation of the invariant subspace problem?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...
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### Is there a proof that the $C^{*}$-algebras don't see the invariant subspace problem?

This post is an appendix of this one. Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Invariant subspace problem: Let $T \in B(H)$. Is ...
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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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### 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 ...
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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 $... 1answer 236 views ### Hodge de Rham operator and orientability Let$(M,g)$be a Riemannian manifold. One can consider the exterior algebra bundle$\Lambda(T^*M)$. The sections of this bundle are differential forms, to be noted by$\Omega^k(M)$. One can consider ... 1answer 157 views ### Convergence of functionals on compact projections on a separable Hilbert space Let$H$be a separable Hilbert space over$\mathbb{C}$, say$\ell_2$for simplicity. Let$\mathcal{K}(H)$denote the space of all compact operators on$H$and$\mathcal{P}(H)$the set of all finite ... 2answers 668 views ### Can one hear the shape of a drum for operators? M. Kac in his famous paper "Can one hear the shape of a drum?" asked whether one can "hear" the area of the ambient domain by looking at the spectral picture. Although he was not the first who came up ... 1answer 362 views ### Existence of a projection operator onto subspace of Hilbert space Let$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$(so finite linear combinitions are dense) which is not ... 1answer 406 views ### What happens if we rotate the kernel of an integral operator? Given an integral operator$K$on$L^2(\mathbb R)$with kernel$k(x, y)$, consider the integral operator$L$on$L^2(\mathbb R)$, whose kernel has the form$k(\alpha x+\beta y, \gamma x+\delta y)$, ... 1answer 134 views ### Simple$Z^{*}$algebra What is an example of a simple$C^{*}$algebra which all elements are (two sided or equivalently one sided) zero divisor? 1answer 55 views ### Operators on Hilbert$C^*$-module and families of Fredholm operators If$A$is a$C^*$-algebra, there is a notion of Hilbert$A$-module (which is something like Hilbert space but the inner product takes values in$A$). The standard example is$H_A:=\{(a_n)_{n=1}^{\...
I would like to find ALL eigenfunctions to the operator, for $f$ a real function on R+*: $f \rightarrow \sum_{1}^{\infty} f(nx)$ So to find $f$ such that: $\sum_{1}^{\infty} f(nx) = \lambda f(x)$ ...
Problem Given a Banach space $E$. Denote compact sets by $\mathcal{C}$, compact operators by $\mathcal{C}(X,Y)$, and finite rank operators by $\mathcal{F}(X,Y)$. Suppose it has the approximation ...