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Local spectrum and Kaplansky's theorem on algebraic operators

100%

Drissi D.

Colloquium Mathematicum

1998

tom 75

nr 2

159-165

Using elementary arguments we improve former results of P. Vrbová concerning local spectrum. As a consequence, we obtain a new proof of Kaplansky’s theorem on algebraic operators on a Banach space.

On solutions of integral equations with analytic kernels and rotations

100%

Nguyen M., Nguyen T.

Annales Polonici Mathematici

1996

tom 63

nr 3

293-300

We deal with a class of integral equations on the unit circle in the complex plane with a regular part and with rotations of the form (*) x(t) + a(t)(Tx)(t) = b(t), where $T = M_{n₁,k₁} ... M_{n_m,k_m}$ and $M_{n_j,k_j}$ are of the form (3) below. We prove that under some assumptions on analytic continuation of the given functions, (*) is a singular integral equation for m odd and is a Fredholm equation for m even. Further, we prove that T is an algebraic operator with characteristic polynomial $P_T(t) = t³ - t$. By means of the Riemann boundary value problems, we give an algebraic method to obtain all solutions of equation (*) in closed form.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Colloquium Mathematicum

1 Drissi D.

1 Nguyen M.

1 Nguyen T.

1 1998

1 1996

Wyniki wyszukiwania

Local spectrum and Kaplansky's theorem on algebraic operators

On solutions of integral equations with analytic kernels and rotations