Inverse-closedness of the set of integral operators with L<sub>1</sub>-continuously varying kernels

Inverse-closedness of the set of integral operators with L₁-continuously varying kernels

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文摘

Let N be an integral operator of the form

(N u) (x) = \int_{R^{c}} n (x, x - y) u (y) d y

acting in L_p(R^c)

L_{p} (R^{c})

with a measurable kernel n satisfying the estimate

|n(x,y)|≤β(y),

| n (x, y) | \leq β (y),

where β∈L₁

β \in L_{1}

. It is proved that if the function b5589e13d0c15e56fe71f606f" title="Click to view the MathML source">t↦n(t,⋅)

t \mapsto n (t, \cdot)

is continuous in the norm of 93bb3f68546ef93bbd4716eb3de" title="Click to view the MathML source">L₁

L_{1}

and the operator 1+N

1 + N

has an inverse, then (1+N)⁻¹=1+M

{(1 + N)}^{- 1} = 1 + M

, where M is an integral operator possessing the same properties.

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