Positive answer to the invariant and hyperinvariant subspaces problems for hyponormal operators

The question whether every operator on infinite-dimensional Hilbert space 𝐻 has a nontrivial invariant subspace or a nontrivial hyperinvariant subspace is one of the most difficult problems in operator theory. This problem is open for more than half a century. A subnormal operator has a nontrivial invariant subspace, but the existence of nontrivial invariant subspace for a hyponormal operator 𝑇 still open. In this paper we give an affirmative answer of the existence of a nontrivial hyperinvariant subspace for a hyponormal operator. More generally, we show that a large classes of operators containing the class of hyponormal operators have nontrivial hyperinvariant subspaces. Finally, every generalized scalar operator on a Banach space 𝑋 has a nontrivial invariant subspace.

Constraints on quasinormal modes and bounds for critical points from pole-skipping

We consider a holographic thermal state and perturb it by a scalar operator whose associated real-time Green’s function has only gapped poles. These gapped poles correspond to the non-hydrodynamic quasinormal modes of a massive scalar perturbation around a Schwarzschild black brane. Relations between pole-skipping points, critical points and quasinormal modes in general emerge when the mass of the scalar and hence the dual operator dimension is varied. First, this novel analysis reveals a relation between the location of a mode in the infinite tower of quasinormal modes and the number of pole-skipping points constraining its dispersion relation at imaginary momenta. Second, for the first time, we consider the radii of convergence of the derivative expansions about the gapped quasinormal modes. These convergence radii turn out to be bounded from above by the set of all pole-skipping points. Furthermore, a transition between two distinct classes of critical points occurs at a particular value for the conformal dimension, implying close relations between critical points and pole-skipping points in one of those two classes. We show numerically that all of our results are also true for gapped modes of vector and tensor operators.

Momentum space spinning correlators and higher spin equations in three dimensions

In this article, we explicitly compute in momentum space the three and four-point correlation functions involving scalar and spinning operators in the free bosonic and the free fermionic theory in three dimensions. We also evaluate the five-point function of the scalar operator in the free bosonic theory. We discuss techniques which are more efficient than the usual PV reduction to evaluate one loop integrals. Our techniques can be easily generalised to momentum space correlators of complicated spinning operators and to higher point functions. The three dimensional fermionic theory has the interesting feature that the scalar operator $$ \overline{\psi}\psi $$ ψ ¯ ψ is odd under parity. To account for this, we develop a parity odd basis which is useful to write correlation functions involving spinning operators and an odd number of $$ \overline{\psi}\psi $$ ψ ¯ ψ operators. We further study higher spin (HS) equations in momentum space which are algebraic in nature and hence simpler than their position space counterparts. We use them to solve for three-point functions involving spinning operators without invoking conformal invariance. However, at the level of four-point functions, solving the HS equation requires additional constraints that come from conformal invariance and we could only verify that our explicit results solve the HS equation.

Gravitational duals to the grand canonical ensemble abhor Cauchy horizons

The gravitational dual to the grand canonical ensemble of a large N holographic theory is a charged black hole. These spacetimes — for example Reissner- Nordström-AdS — can have Cauchy horizons that render the classical gravitational dynamics of the black hole interior incomplete. We show that a (spatially uniform) deformation of the CFT by a neutral scalar operator generically leads to a black hole with no inner horizon. There is instead a spacelike Kasner singularity in the interior. For relevant deformations, Cauchy horizons never form. For certain irrelevant deformations, Cauchy horizons can exist at one specific temperature. We show that the scalar field triggers a rapid collapse of the Einstein-Rosen bridge at the would-be Cauchy horizon. Finally, we make some observations on the interior of charged dilatonic black holes where the Kasner exponent at the singularity exhibits an attractor mechanism in the low temperature limit.

Three ways to solve critical ϕ4 theory on 4 − 𝜖 dimensional real projective space: Perturbation, bootstrap, and Schwinger–Dyson equation

In this paper, we solve the two-point function of the lowest dimensional scalar operator in the critical [Formula: see text] theory on [Formula: see text] dimensional real projective space in three different methods. The first is to use the conventional perturbation theory, and the second is to impose the cross-cap bootstrap equation, and the third is to solve the Schwinger–Dyson equation under the assumption of conformal invariance. We find that the three methods lead to mutually consistent results but each has its own advantage.

Analytical studies on holographic superconductor in the probe limit

We investigate the holographic superconductor model constructed in the (2[Formula: see text]+[Formula: see text]1)-dimensional AdS soliton background in the probe limit. With analytical methods, we obtain the formula of critical phase transition points with respect to the scalar mass. We also generalize this formula to higher-dimensional space–time. We mention that these formulas are precise compared to numerical results. In addition, we find a correspondence between the value of the charged scalar field at the tip and the scalar operator at infinity around the phase transition points.

Multiple solutions for an elliptic system with indefinite Robin boundary conditions

Multiplicity of solutions is proved for an elliptic system with an indefinite Robin boundary condition under an assumption that links the linearisation at 0 and the eigenvalues of the associated linear scalar operator. Our result is based on a precise calculation of the topological degree of a suitable fixed point operator over large and small balls.

Para colmo, scalar operator and additive connector

This study addresses the development undergone by para colmo, a form which is in the process of becoming fixed as an argumentative operator and additive connector. Para colmo has undergone a process of subjectification (Traugott 1995a), which is crucial in the development of discourse markers, where the presence of the speaker is manifested in various ways to cover important areas of meaning in the interpretation of the discourse. It has polarized its meaning in the negative and has progressed to indicate scalar saturation, or a position at the highest level of the argumentative scale, with an evaluation of excess on the part of the speaker. The study has been carried out on the basis of both diachronic and synchronic corpora compiled by the Real Academia Española, and covers the period from 1200 to 2004.