Geodesic
On Unbounded Volterra Operators
Canadian journal of mathematics, Tome 36 (1984) no. 5, pp. 914-923

Voir la notice de l'article provenant de la source Cambridge University Press

Similarity invariants of certain families of Volterra operators acting in Lp [0, 1] were first determined by Kalisch in [4] and [5]. Subsequently, Kantorovitz made an extensive study of perturbations of the form Tα = M + αN, where M is the operation of multiplication by x, and N is the Volterra operator both acting in Lp [0, 1], and determined that Re α is a similarity invariant for the class of Tα's (cf. [7], and earlier work cited there). This result was eventually generalized to unbounded M (cf. [6]), and one direction was proved in [2] for both M and N unbounded: if Re α = Re β, then Tα is similar to Tβ. Properties of the Riemann-Liouville semigroup of fractional integration play a key role, for in all cases the ensuing similarity turns out to be implemented by N(i(lm(α — β))), where {N(iγ)} is the boundary group of the semigroup of fractional integration. 
Hughes, Rhonda J. On Unbounded Volterra Operators. Canadian journal of mathematics, Tome 36 (1984) no. 5, pp. 914-923. doi: 10.4153/CJM-1984-052-2
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[1] 1. Hille, E. and Phillips, R. S., Functional analysis and semigroups, rev. éd., Amer. Math. Soc. Colloq Publ. 31 (Providence, R.I., 1957). Google Scholar

[2] 2. Hughes, R. J. and Kantorovitz, S., Boundary values of holomorphic semigroups of unbounded operators and similarity of certain perturbations, J. Functional Analysis 29 (1978), 253–273. Google Scholar

[3] 3. Hughes, R. J., The approximation of fractional powers of a closed operator, Illinois J. Math. 24 (1980), 333–343. Google Scholar

[4] 4. Kalisch, G. K., On similarity, reducing manifolds, and unitary equivalence of certain Volterra operators, Ann. of Math. 66 (1957), 481–494. Google Scholar

[5] 5. Kalisch, G. K., On similarity invariants of certain operators in L, Pacific J. Math. 11 (1961), 247–252. Google Scholar

[6] 6. Kantorovitz, S. and Pei, K. J., Pairs of operators satisfying the Volterra commutation relation, Indiana Univ. Math. J. 23 (1974), 1177–1197. Google Scholar

[7] 7. Kantorovitz, S., Commutators, Ck-classification, and similarity of operators, Trans. Amer. Math Soc. 156 (1971), 193–218. Google Scholar

[8] 8. Kato, T., Perturbation theory for linear operators (Springer-Verlag, New York, 1966). Google Scholar

[9] 9. Okikiolu, G. O., Aspects of the theory of bounded integral operators in LP-spaces (Academic Press, London, 1971). Google Scholar

[10] 10. Reed, M. and Simon, B., Methods of modern mathematical physics II: Fourier analysis, selfadjointness (Academic Press, New York, 1975). Google Scholar

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