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We study resonances for Jacobi operators on the half lattice with matrix-valued coefficients and finitely supported perturbations. We describe a forbidden domain, the geometry of resonances and their asymptotics when the main coefficient of the perturbation (determining its length of the support) goes to zero. Moreover, we show that

In this paper, we investigate the asymptotic behavior as infinity of a class of degenerate Monge–Ampère equations in half spaces. Meanwhile, we establish the Schauder estimates on a class of degenerate linear elliptic equations.

We study the behaviour of functions of self-adjoint operators under relatively bounded and relatively trace class perturbation. We introduce and study the class of relatively operator Lipschitz functions. An essential role is played by double operator integrals. We also consider the class of resolvent Lipschitz functions. Then we obtain a trace formula in the case of relatively trace class perturbations and show that the maximal class of function for which the trace formula holds in the case of relatively trace class perturbations coincides with the class of relatively operator Lipschitz functions. Our methods also give us a new approach to the inequality for the spectral shift function $��\xi �$ in the case of relatively trace class perturbations.

The following problems related to almost complex structures on a twistor space are discussed: (1) when two almost complex structures defined by a morphism of the twistor space and two conformal metrics on the base manifold coincide; (2) when the Atiyah–Hitchin–Singer or Eells–Salamon almost complex structures on the twistor space defined by two metric connections on the base Riemannian manifold coincide.

This paper is concerned with a class of bounded pesudoconvex Hartogs domains, which is defined by the inequality

For two variable Lebesgue spaces $�{\ell }_{p_n}$ and $�{\ell }_{q_n}$, with , we provide necessary and sufficient conditions under which the natural embeddings $��i�d�:�{\ell }_{p_n}�\to �{\ell }_{q_n}�$ are strictly or finitely strictly singular. We also provide estimates for the Bernstein numbers of the natural embedding $��i�d�$ and show how they depend on the exponents $�p_n$ and $�{}_{}$

Suppose that $�K$ and $�S$ are locally compact Hausdorff spaces and $�X$ is a Hilbert space. It is proven that if there exist real numbers $��M�\ge �1�$, $��L�\ge �0�$ and a map $�T$ from $��C_0��(�K�,�X�)��$ to $��C_0��(�S�,�X�)��$ satisfying

In this case, as an immediate consequence, $�\phi$ generates a linear isometry of $��C_0��\left(�K�\right)��$ into