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Local spectral theory of endomorphisms of the disk algebra

Trivedi S., Chandra H.

Demonstratio Mathematica

2016

Vol. 49, nr 1

95--106

Let A(D) denote the disk algebra. Every endomorphism of A(D) is induced by some φ (…) A(D) with ║φ║ ≤ 1. In this paper, it is shown that if φ is not an automorphism of D and φ has a fixed point in the open unit disk then the endomorphism induced by φ is decomposable if and only if the fixed set of φ is singleton. Further, we determine the local spectra of the endomorphism induced by φ in the cases when the fixed set of φ either includes unit circle or is a singleton.

On the Nemytskii operator in the space of functions of bounded (p, 2, α)-variation with respect to the weight function

Aziz W.

Demonstratio Mathematica

2014

Vol. 47, nr 4

933--948

In this paper, we consider the Nemytskii operator (H f) (t) = h(t, f(t)), generated by a given function h. It is shown that if H is globally Lipschitzian and maps the space of functions of bounded (p, 2, α)-variation (with respect to a weight function α) into the space of functions of bounded (q, 2, α)-variation (with respect to α) 1 < q < p, then H is of the form (H f) (t) = A(t)f(t) + B(t). On the other hand, if 1 < p < q then H is constant. It generalize several earlier results of this type due to Matkowski–Merentes and Merentes. Also, we will prove that if a uniformly continuous Nemytskii operator maps a space of bounded variation with weight function in the sense of Merentes into another space of the same type, its generator function is an affine function.

Injective composition operators on Lorentz-Bochner spaces

Datt G., Arora S. C.

Demonstratio Mathematica

2013

Vol. 46, nr 4

741--756

In this paper, we extend the notion of essential range to vector-valued functions and present various equivalent conditions for the injectiveness of the composition operators alongwith a characterisation for measurable transformations inducing composition operators between Lorentz-Bochner spaces. Some aspects of the weighted composition operators on Lorentz-Bochner spaces, induced by a measurable transformation and an operator valued map, are also discussed.

Generalized integration operators from Bloch type spaces to weighted BMOA spaces

Sharma S. D., Sharmat A.

Demonstratio Mathematica

2011

Vol. 44, nr 2

373--390

In this paper, we consider a generalized integration operator (…) induced by holomorphic maps g and φ of the open unit disk D, where φ(D) ⊂ D and n is a positive integer. We characterize boundedness and compactness of (…) from Bloch type spaces to weighted BMOA spaces by using logarithmic Carleson measure characterization of the weighted BMOA spaces.

Ascent and descent of composition operators on lp spaces

Chandra H., Kumar P.

Demonstratio Mathematica

2010

Vol. 43, nr 1

161-165

Let lp (1 < p