Universal Continuous Calculus for Su*-Algebras

具有多项式增长连续函数演算的格有序代数

Lattice-ordered algebras admitting a polynomial growth continuous function calculus

本文研究了具有恒等元的阿基米德格有序代数中多项式增长连续函数演算的存在性，通过同态映射等价于特定子代数的完备性条件，揭示了向量格的正齐次连续函数演算特性。

On functional calculus for Hermitian elements of Banach algebras: the norm and spectral radius

Let $ A$ be a complex unital Banach algebra. An element $a \in A$ is said to be Hermitian, if $ \| \exp (ita) \| =1$ for all $t\in R$. In the case of the algebra of bounded linear operators in a Hilbert space this Hermitian property agrees with the ordinary selfadjointness. If $a \in A$ is Hermitian, then $|a|=||a||$, where $|a|$ denotes the spectral radius of $a$. A function $F: R\to \C$ is called the universal symbol if $ \|| F(a)||= |F(a)|$\ for each $ A$ and all Hermitian $a\in A$. We characterize universal symbols in terms of positive definite functions.

连续函数格中的离散停时

Discrete stopping times in the lattice of continuous functions

本文研究了连续函数格中的泛函演算与离散停时，证明了Daniell泛函演算可表示为函数的逐点复合，并推导出与概率论中相类似的停时与停时过程表示。

Joint functional calculus in algebra of polynomial tempered distributions

In this paper we develop a functional calculus for a countable system of generators of contraction strongly continuous semigroups. As a symbol class of such calculus we use the algebra of polynomial tempered distributions. We prove a differential property of constructed calculus and describe its image with the help of the commutant of polynomial shift semigroup. As an application, we consider a function of countable set of second derivative operators.

Polynomial as a new variable - a Banach algebra with a functional calculus

Given any square matrix or a bounded operator $A$ in a Hilbert space such that $p(A)$ is normal (or similar to normal), we construct a Banach algebra, depending on the polynomial $p$, for which a simple functional calculus holds. When the polynomial is of degree $d$, then the algebra deals with continuous $\mathbb C^d$-valued functions, defined on the spectrum of $p(A)$. In particular, the calculus provides a natural approach to deal with nontrivial Jordan blocks and one does not need differentiability at such eigenvalues.

Construction of a tensor product algebra with a multicentric functional calculus

In the multicentric calculus one takes a polynomial with simple roots as a new global variable and replaces scalar functions φ by functions f taking values in C^d with d the degree of the polynomial leading to an efficient holomorphic functional calculus for bounded operators. This calculus was extended for non-holomorphic functions, so that if A is not diagonalizable one can find a p such that p(A) is diagonalizable and apply the calculus to all matrices. This was done in [7] by creating a Banach algebra for C^d-valued continuous functions in such a way that the original functions φ appear as…

Functional calculus for diagonalizable matrices

For an arbitrary function f:Ω\rightarrow C (where Ωis a subset of the field C) and a positive integer k let f act on all diagonalizable complex matrices whose all eigenvalues lie in Omega in the following way: f[P Diag(z1,...,zk) P-1] = P Diag(f(z1),...,f(zk)) P-1 for arbitrary numbers z1,...,zk in Ωand an invertible k \times k matrix P. The aim of the paper is to fully answer the question of when the function fop defined above is continuous for fixed k. In particular, it is shown that if Ωis open in C, then fop is continuous for fixed k > 2 iff f is holomorphic; and if Ωis an interval in R an…

A Loomis-Sikorski theorem and functional calculus for a generalized Hermitian algebra

A generalized Hermitian (GH-) algebra is a generalization of the partially ordered Jordan algebra of all Hermitian operators on a Hilbert space. We introduce the notion of a gh-tribe, which is a commutative GH-algebra of functions on a nonempty set $X$ with pointwise partial order and operations, and we prove that every commutative GH-algebra is the image of a gh-tribe under a surjective GH-morphism. Using this result, we prove each element $a$ of a GH-algebra $A$ corresponds to a real observable $ξ_a$ on the $σ$-orthomodular lattice of projections in $A$ and that $ξ_a$ determines the spectral…

Functional Itô calculus and stochastic integral representation of martingales

We develop a nonanticipative calculus for functionals of a continuous semimartingale, using an extension of the Ito formula to path-dependent functionals which possess certain directional derivatives. The construction is based on a pathwise derivative, introduced by Dupire, for functionals on the space of right-continuous functions with left limits. We show that this functional derivative admits a suitable extension to the space of square-integrable martingales. This extension defines a weak derivative which is shown to be the inverse of the Ito integral and which may be viewed as a nonanticip…

Banach空间中的光滑函数演算与谱定理

Smooth Functional Calculus and Spectral Theorem in Banach Spaces

本文通过投影族将谱定理推广至Banach代数与Banach空间算子，利用Cauchy-Pompeiu公式建立光滑及连续函数演算，并证明其为Hilbert空间情形的适当推广。