Annals of Functional Analysis最新文献

In this article, we study the multiple solutions of a class of variable-order Schrödinger–Kirchhoff-type double-phase system, the equation has a nonlinear term of the concave–convex nonlinearities with variable exponent and a new type critical term which is better suitable for double-phase problem. Using the concentration-compactness principle and Kajikiya’s symmetric mountain pass theorem, the existence of infinitely many solutions for suitable small parameters (mu) and (nu _i,i=1,2) has been obtained, respectively. This implies that infinite solutions exist when the parameters (mu) and (max { nu _1,nu _2}) lie within an (mathbb {L})-shaped region (see Fig. 1). A technique is developed to determine the geometry of energy functionals in such Schrödinger–Kirchhoff-type systems with concave–convex terms and variable exponents.

In this paper, we investigate the notions of multiplicative and ternary domains for completely positive (CP) maps between pro-(C^*)-algebras, and establish a Schwarz-like inequality for such maps which are contractive. Along with this, we study the (phi )-module domain and ternary domain for a (phi )-map (Phi ), where (Phi ) is a CP-map between two Hilbert pro-(C^*)-modules. Through a detailed construction, we demonstrate that the ternary domain of a (phi )-map (Phi ) coincides with the (phi )-module domain of (Phi ). Furthermore, we establish relationships between the multiplicative and ternary domains of a CP-map and the associated Stinespring triple. In addition, we derive connections between the Stinespring-like representation for (phi )-maps and the (phi )-module domain of such maps.

This note aims to introduce an equivalent norm for variable Herz spaces. While an extrapolation result is already known for these spaces, the equivalent norm proposed in this paper provides an example that extends beyond the scope of the existing extrapolation result

We characterize the Schatten class Toeplitz operators associated with a positive Borel measure on the weighted harmonic Bergman space (L_{h,omega }^2(mathbb {D})) over the unit disk. Furthermore, we establish the necessary and sufficient condition for the Toeplitz operators on (L_{h,omega }^2(mathbb {D})) belonging to the Schatten (hbar)-class, where (hbar) is defined as a continuous increasing convex function.

There has been significant developments in the classification of boundary conditions of positive symmetric systems, also known as Friedrichs systems, after the introduction of operator theoretic framework. We take a step forward towards applying the abstract theory to the classical framework by studying Friedrichs systems on an interval. Dealing with some difficulties related to the smoothness of eigenvectors, here we present an explicit expression for the dimensions of the kernels of Friedrichs operators only in terms of the values of the coefficients at the end-points of the interval. In particular, this allows for a characterisation of all admissible boundary conditions, i.e. those leading to bijective realisations.

We use spectral flow to present a new proof of Levinson’s theorem for Schrödinger operators on (mathbb {R}^n) with smooth compactly supported potential. Our proof is valid in all dimensions and in the presence of resonances. The statement is expressed in terms of the spectral shift function and the “high energy corrected time delay” following Guillopé and others.

We build on a characterization of inner functions f due to Le, in terms of the spectral properties of the operator (V=M_f^*M_f) and study to what extent the cyclicity on weighted Hardy spaces (H^2_omega ) of the function (z mapsto a-z) can be similarly inferred from the spectral properties of the corresponding operator V. We describe several properties of the spectra that hold in a large class of spaces and then, we focus on the particular case of Bergman-type spaces, for which we describe completely the spectrum of such operators and find all eigenfunctions.

This paper investigates the dilation problem for oblique dual pairs of g-frame sequences in Hilbert spaces. It is demonstrated that any oblique dual pair (Type I dual pair) of g-frame sequences in a Hilbert space can be dilated to an oblique dual pair (Type I dual pair) of g-Riesz sequences in a larger Hilbert space. Furthermore, a characterization is established for a g-frame sequence to possess a Parseval Type II dual through orthogonal dilation. Finally, a condition is provided under which an oblique dual pair of g-frame sequences in a Hilbert space can be obliquely dilated to a dual pair of g-Riesz bases for the same space.

We study the normalized solutions of the (L^2)-critical Schrödinger–Poisson system with an external potential (V(x)=|x|^2) in ({mathbb {R}}^2), which can be described by the constraint minimization problem. When the magnetic field is attractive, we prove that there is a threshold (a^*in (0,infty )) such that the constraint minimizer exists if and only if the interaction strength (a<a^*). Moreover, for the repulsive case, there exists a minimizer if (a<a^*), while there does not exist any minimizer if (a>a^*). Particularly, after analyzing its limiting behavior, we then obtain the uniqueness of positive minimizers as (anearrow a^*) by overcoming the sign-changing property of the logarithmic convolution and the non-invariance under translations of the harmonic potential.

In this paper, we study the boundedness of global continuous linear operators on smooth manifolds. Using the notion of a global symbol, we extend a classical condition of Hörmander type to guarantee the (L^p)-(L^q)-boundedness of global operators. Our approach links the mapping properties of continuous linear operators on smooth manifolds with the (L^p)-estimates of eigenfunctions of operators including a variety of examples, harmonic oscillators, anharmonic oscillators, etc. First, we investigate (L^p)-boundedness of pseudo-multipliers in the setting of Hörmander–Mihlin type conditions. We also prove (L^infty)-BMO estimates for pseudo-multipliers. Later, we concentrate our investigation to settle (L^p)-(L^q) boundedness of the Fourier multipliers and pseudo-multipliers operators for the range (1<p le 2 le q<infty .) On the way to achieve our goal of (L^p)-(L^q) boundedness, we prove two classical inequalities, namely, Paley inequality and Hausdorff–Young–Paley inequality for smooth manifolds. Finally, we present some examples about the well-posedness of abstract non-linear equations.