Volterra operator norms : a brief survey

Thomas Ransford

doi:10.2478/mjpaa-2023-0018

.blurhash-client-img { display: none !important; }

Volterra operator norms : a brief survey

Moroccan Journal of Pure and Applied Analysis

Volume 9 (2023): Issue 2 (May 2023)

By: Thomas Ransford

Open Access

|Jun 2023

Abstract

In this expository article, we discuss the evaluation and estimation of the operator norms of various functions of the Volterra operator.

References

Allan, G. R., Power-bounded elements and radical Banach algebras. Linear operators (Warsaw, 1994), 38, 9–16,Polish Acad. Sci. Inst. Math., Warsaw, 1997.
Search in Google Scholar Back to article
Böttcher, A. and Dörfler, P., On the best constants in inequalities of the Markov and Wirtinger types for polynomials on the half-line. Linear Algebra Appl.Vol 430, N 4,1057–1069,2009.
Search in Google Scholar Back to article
Böttcher, A. and Dörfler, P., Weighted Markov-type inequalities, norms of Volterra operators, and zeros of Bessel functions.Math. Nachr. Vol. 283, No. 1,40–57, 2010.
Search in Google Scholar Back to article
Böttcher, A. and Widom, H., On the eigenvalues of certain canonical higher-order ordinary differential operators. J. Math. Anal. Appl. Vol. 322, No. 2, 990–1000, 2006.
Search in Google Scholar Back to article
Crouzeix, M., Bounds for analytical functions of matrices. Integral Equations Operator Theory. Vol. 48, No. 4, 461–477, 2004.
Search in Google Scholar Back to article
Crouzeix, M. and Palencia, C., The numerical range is a(1+2) a\left( {1 + \sqrt 2 } \right) -spectral set. SIAM J. Matrix Anal. Appl. Vol. 38,No. 2, 649–655, 2017.
Search in Google Scholar Back to article
Franco, Z. M. and Kaper, H. G. and Kwong, M. K. and Zettl, A., Best constants in norm inequalities for derivatives on ahalf-line. Proc. Roy. Soc. Edinburgh Sect. A. Vol. 100, No. 1-2, 67–84, 1985.
Search in Google Scholar Back to article
Halmos, P. R., A Hilbert space problem book. Encyclopedia of Mathematics and its Applications. Vol. 17, Edition Second, 1982.
Search in Google Scholar Back to article
Horgan, C. O., A note on a class of integral inequalities. Proc. Cambridge Philos. Soc.Vol. 74, 127–131, 1973.
Search in Google Scholar Back to article
Kershaw, D., Operator norms of powers of the Volterra operator. J. Integral Equations Appl. Vol. 11, No. 3, 351–362, 1999.
Search in Google Scholar Back to article
Khadkhuu, L. and Tsedenbayar, D., A note about Volterra operator. Math. Slovaca. Vol. 68, No. 5, 2018.
Search in Google Scholar Back to article
Khadkhuu, L. and Tsedenbayar, D., On the numerical range and numerical radius of the Volterra operator. Izv. Irkutsk. Gos. Univ. Ser. Mat. Vol. 24, pp. 102–108, 2018.
Search in Google Scholar Back to article
Lao, N. and Whitley, R., Norms of powers of the Volterra operator. Integral Equations Operator Theory. Vol. 27, No. 4, 419–425,1997.
Search in Google Scholar Back to article
Little, G. and Reade, J. B., Estimates for the norm of the nth indefinite integral. Bull. London Math. Soc. Vol. 30, No. 5, 539–542, 1998.
Search in Google Scholar Back to article
Lumer, G., Semi-inner-product spaces. Trans. Amer. Math. Soc. Vol. 100, 29–43, 1961.
Search in Google Scholar Back to article
Lyubich, Y. and Tsedenbayar, D., The norms and singular numbers of polynomials of the classical Volterra operator in L2(0, 1). Studia Math. Vol. 199, No. 2, 171–184, 2010.
Search in Google Scholar Back to article
Ransford, T. and Walsh, N., Norms of polynomials of the Volterra operator, J. Math. Anal. Appl. 517 (2023), paper no. 126626.
Search in Google Scholar Back to article
Ter Elst, A. F. M. and Zem´anek, J., Contractive polynomials of the Volterra operator. Studia Math. Vol. 240, No. 3, 201–211, 2018.
Search in Google Scholar Back to article
Thorpe, B., The norm of powers of the indefinite integral operator on (0, 1). Bull. London Math. Soc. Vol. 30, No. 5, 543–548, 1998.
Search in Google Scholar Back to article