On the Spectral Properties of Selfadjoint Partial Integral Operators with a Nondegenerate Kernel

We consider bounded selfadjoint linear integral operators $ T_{1} $ and $ T_{2} $ in the Hilbert space $ L_{2}([a,b]\times[c,d]) $ which are usually called partial integral operators. We assume that $ T_{1} $ acts on a function $ f(x,y) $ in the first argument and performs integration in $ x $ , while $ T_{2} $ acts on $ f(x,y) $ in the second argument and performs integration in $ y $ . We assume further that $ T_{1} $ and $ T_{2} $ are bounded but not compact, whereas $ T_{1}T_{2} $ is compact and $ T_{1}T_{2}=T_{2}T_{1} $ . Partial integral operators arise in various areas of mechanics, the theory of integro-differential equations, and the theory of Schrödinger operators. We study the spectral properties of $ T_{1} $ , $ T_{2} $ , and $ T_{1}+T_{2} $ with nondegenerate kernels and established some formula for the essential spectra of $ T_{1} $ and $ T_{2} $ . Furthermore, we demonstrate that the discrete spectra of $ T_{1} $ and $ T_{2} $ are empty, and prove a theorem on the structure of the essential spectrum of $ T_{1}+T_{2} $ . Also, under study is the problem of existence of countably many eigenvalues in the discrete spectrum of $ T_{1}+T_{2} $ .

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