Toeplitz Operators whose Symbols Are Borel Measures

In this paper, we are concerned with Toeplitz operators whose symbols are complex Borel measures. When a complex Borel measure μ on the unit circle is given, we give a formal definition of a Toeplitz operator T μ with symbol μ , as an unbounded linear operator on the Hardy space. We then study various properties of T μ . Among them, there is a theorem that the domain of T μ is represented by a trichotomy. Also, it was shown that if the domain of T μ contains at least one polynomial, then T μ is densely defined. In addition, we give evidence for the conjecture that T μ with a singular measure μ reduces to a trivial linear operator.

Feynman Integral and a Change of Scale Formula about the First Variation and a Fourier–Stieltjes Transform

We prove that the Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation of F(x)=exp{∫0Tθ(t,x(t))dt} successfully exist under the certain condition, where θ(t,u)=∫Rexp{iuv}dσt(v) is a Fourier–Stieltjes transform of a complex Borel measure σt∈M(R) and M(R) is a set of complex Borel measures defined on R. We will find this condition. Moreover, we prove that the change of scale formula for Wiener integrals about the first variation of F(x) sucessfully holds on the Wiener space.

Conditional Fourier-Feynman Transforms with Drift on a Function Space

In this paper we derive change of scale formulas for conditional analytic Fourier-Feynman transforms and conditional convolution products of the functions which are the products of generalized cylinder functions and the functions in a Banach algebra which is the space of generalized Fourier transforms of the complex Borel measures on L2[0,T] using two simple formulas for conditional expectations with a drift on an analogue of Wiener space. Then we prove that the conditional transform of the conditional convolution product can be expressed by the product of the conditional transforms of each function. Finally we establish various changes of scale formulas for the conditional transforms and the conditional convolution products.

CONDITIONAL FEYNMAN INTEGRAL AND SCHRÖDINGER INTEGRAL EQUATION ON A FUNCTION SPACE

Let Cr[0,t] be the function space of the vector-valued continuous paths x:[0,t]→ℝr and define Xt:Cr[0,t]→ℝ(n+1)r by Xt(x)=(x(0),x(t1),…,x(tn)), where 0<t1<⋯<tn=t. In this paper, using a simple formula for the conditional expectations of the functions on Cr[0,t] given Xt, we evaluate the conditional analytic Feynman integral Eanfq[Ft∣Xt] of Ft given by where θ(s,⋅) are the Fourier–Stieltjes transforms of the complex Borel measures on ℝr, and provide an inversion formula for Eanfq[Ft∣Xt]. Then we present an existence theorem for the solution of an integral equation including the integral equation which is formally equivalent to the Schrödinger differential equation. We show that the solution can be expressed by Eanfq[Ft∣Xt] and a probability distribution on ℝr when Xt(x)=(x(0),x(t)).

Absolutely continuous measures and compact composition operator on spaces of Cauchy transforms

The analytic self-map of the unit diskD,φis said to induce a composition operatorCφfrom the Banach spaceXto the Banach spaceYifCφ(f)=f∘φ∈Yfor allf∈X. Forz∈Dandα>0, the families of weighted Cauchy transformsFαare defined byf(z)=∫TKxα(z)dμ(x), whereμ(x)is complex Borel measure,xbelongs to the unit circleT, and the kernelKx(z)=(1−x¯z)−1. In this paper, we will explore the relationship between the compactness of the composition operatorCφacting onFαand the complex Borel measuresμ(x).