## The measure of non-compactness and approximation numbers of certain Volterra integral operators.(English)Zbl 0788.45013

This paper studies Volterra integral operators $$K:L^ p(\mathbb{R}^ +) \to L^ q (\mathbb{R}^ +)$$ of the form $Kf(x)=v(x) \int^ x_ 0k(x,y) u(y)f(y) dy,$ where $$1<p \leq q<\infty$$ and $$k$$, $$u$$ and $$v$$ are prescribed functions. Under appropriate conditions we give sharp upper and lower estimates of the distance of $$K$$ from the compact operators; and when $$K$$ is compact, we provide upper and lower estimates for its approximation numbers when $$p=q =2$$ and $$k(x,y)=p_ n(x-y)$$, where $$p_ n$$ is a polynomial of degree $$n$$.
As an example we show that when $$u(x)=e^{Dx}$$ and $$v(x)=e^{-Bx}$$, with $$0<D<B$$, and $$k(x,y)=x-y$$, then the $$m$$th-approximation number of $$K:L^ 2(\mathbb{R}^ +) \to L^ 2(\mathbb{R}^ +)$$ is bounded above and below by multiples of $$m^{-2}$$.

### MSC:

 45P05 Integral operators 45G10 Other nonlinear integral equations 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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### References:

 [1] Bloom, S., Kerman, R.: Weighted norm inequalities for operators of Hardy type. Proc. Am. Math. Soc.113, 135-141 (1991) · Zbl 0753.42010 [2] Canavati, J.A., Galaz-Fontes, F.: Compactness of imbeddings between Banach spaces and applications to Sobolev spaces. J. Lond. Math. Soc., II. Ser.41, 511-525 (1990) · Zbl 0737.46015 [3] Carl, B.: Inequalities of Bernstein-Jackson type and the degree of compactness of operators in Banach spaces. Ann. Inst. Fourier35, 79-118 (1985) · Zbl 0564.47009 [4] Edmunds, D.E., Evans, W.D.: Spectral theory and differential operators. Oxford: Oxford University Press 1987 · Zbl 0628.47017 [5] Edmunds, D.E., Evans, W.D., Harris, D.J.: Approximation numbers of certain Volterra integral operators. J. Lond. Math. Soc., II. Ser.37, 471-489 (1988) · Zbl 0658.47049 [6] Evans, W.D., Harris, D.J.: Sobolev embeddings for generalized ridged domains. Proc. Lond. Math. Soc., III. Ser.54, 141-175 (1987) · Zbl 0591.46027 [7] Juberg, R.K.: Measure of non-compactness and interpolation of compactness for a class of integral transformations. Duke Math. J.41, 511-525 (1974) · Zbl 0291.47027 [8] K?nig, H.: Eigenvalue distribution of compact operators. Basel Boston Stuttgart: Birkh?user 1986 [9] Muckenhaupt, B.: Hardy inequalities with weights. Stud. Math.44, 31-38 (1972) [10] Mynbaev, K.T., Otelbaev, M.O.: Weighted function spaces and the spectrum of differential operators (Russian) Moscow: Nauka 1988 · Zbl 0651.46037 [11] Oinarov, R.: Weighted inequalities for a class of integral operators. Sov. Math., Dokl.44, 291-293 (1992) [12] Pietsch, A.: Operator ideals, (VEB Deutscher Verlag Wiss., Berlin, 1978). · Zbl 0399.47039 [13] Stepanov, V.D.: Weighted inequalities for a class of Volterra convolution operators. J. Lond. Math. Soc., II. Ser.45, 232-242 (1992) · Zbl 0703.42011 [14] Stepanov, V.D.: Weighted norm inequalities of Hardy type for a class of integral operators. Preprint, Institute for Applied Math., Far-Eastern Branch Russian Acad. Sci., Khabarovsk, 1992 [15] Stuart, C.A.: The measure of non-compactness of some linear integral operators. Proc. R. Soc. Edinb., Sect. A71, 167-179 (1973) · Zbl 0314.47029
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