2018年度秋季総合分科会(於:岡山大学)

SUMMARY: In this talk, I will review recent studies on the Hubbard model, which is expected to describe metallic ferromagnetism in strongly correlated electron systems. I will begin with a description of the physical background, then I will present a list of open problems in this field.

As an attempt to solve the problems, a new viewpoint of universality is introduced in strongly correlated electron systems. Our description relies on the operator theoretic correlation inequalities. I explain the Marshall–Lieb–Mattis theorem and Lieb’s theorem from a viewpoint of universality. In addition, from the new perspective, I prove that Lieb’s theorem still holds true even if the electron-phonon and electron-photon interactions are taken into account. I also study Nagaoka–Thouless’ theorem and its stabilities in terms of universality.

SUMMARY: To compute the \(b\)-function of the determinant \(\det (x_{ij})\) is one of the purposes that the Capelli identity was established. Since then a deep relation with the representation theory of Lie group has been understood, and finally the abstract Capelli problem was proposed and settled completely in the paper of Howe and Umeda in 1991. The abstract Capelli problem is a problem asking if every invariant differential operator on \(V\) comes from \(\mathfrak {g}\) or not, where \(\mathfrak {g}\) is a Lie algebra, and \(V\) is a representation space of a finite-dimensional representation of \(\mathfrak {g}\).

If the abstract Capelli problem holds for a prehomogeneous vector space, then the \(b\)-function can be computed easily by using invariant differential operators. But the abstract Capelli problem fails for most of the prehomogeneous vector spaces. Even in such a case there may be a “Capelli identity”, and the \(b\)-function is computed by using it. Thus it is still important to study Capelli identities for prehomogeneous vector spaces where the abstract Capelli problem does not hold.

In this talk I introduce such Capelli identities for some prehomogeneous vector spaces, that is, Capelli identities of “odd” type, which help compute the \(b\)-functions of prehomogeneous vector spaces associated with quivers, Capelli identities for the prehomogeneous vector spaces of parabolic type, which are generalization of those for quivers, and Capelli identities with zero entries, which are related to non-reductive prehomogeneous vector spaces.

SUMMARY: Let \((M, \| \cdot \|_M)\) and \((N, \| \cdot \|_N)\) be normed linear spaces. A mapping \(S \colon M \to N\) is an isometry if and only if \[\| S(f) - S(g) \|_N = \| f - g \|_M\] holds for all \(f, g \in M\). Here, we note that isometries need not be linear maps. If an isometry is surjective, then we see that it is essentially real linear by the Mazur–Ulam theorem. Even if surjective isometries are real linear, they need not be complex linear nor conjugate linear. For example, let \(\mathbb T\) be the unit circle in the complex number field. Then the complex linear space of all linear functions \(az+b\) on \(\mathbb T\) with the supremum norm on \(\mathbb T\) has a surjective real linear isometry that maps \(az+b\) to \(az+\bar {b}\), where \(\bar {b}\) denotes the complex conjugate of \(b\). It is neither complex linear nor conjugate linear isometry. To the best of my knowledge, the structure of surjective isometries on function spaces remains obscure.

To investigate surjective isometries on function spaces, we first examine \(C^1\) space of all continuously differentiable functions defined on the closed unit interval \([0,1]\) with respect to several norms. We also show the structure of surjective isometries on a Banach space of analytic functions on the open unit disc.

SUMMARY: In this talk, I shall introduce my studies and ideas about non-self-adjoint Hamiltonian and some physical operators constructed from biorthogonal sequences in a Hilbert space. The notion of generalized Riesz system plays an important rule such studies. From this reason, I introduce under what assumptions a biorthogonal sequence is a generalized Riesz system and construct some well-defined physical operators.

SUMMARY: In this talk, I shall introduce my studies about relationships between generalized Riesz systems and \(D\)-quasi bases. The notion of \(D\)-quasi bases is an important rule for constructing non-self-adjoint hamiltonian and physical operators in case that \(D_\varphi \) and \(D_\psi \) are not dense in \(H\).

SUMMARY: In this presentation, we will study two approaches for the reducibility of linear ordinary differential equations: algebraic reducibility and the Lyapunov reducibility. For those aspects, several criteria of reducibility will be introduced and discussed. Especially, the generalised Shemesh criterion can be used to check the existence of a common eigenspace, so it can determine if a Fedorov type solution exists.

SUMMARY: We give an operator-theoretical proof of the statement that the transition from a normal conducting state to a superconducting state is a second-order phase transition in the BCS-Bogoliubov model of superconductivity. Here we have no magnetic fields.

SUMMARY: We consider 1-dimensional quantum walks on the line. It is known that if a coin operator rapidly converges to a unitary matrix, then the wave operator exists. We consider the coin operator which slowly converges to a unitary matrix. It is shown that the wave operator does not exist.

SUMMARY: We derive a detection method of edge defects for position-dependent quantum walks using eigenvalues embedded in the continuous spectrum of time evolution operators. The localization occurs if the initial state has an overlap with an eigenfunction of the time evolution operator. However, we cannot detect the existence of edge defects by the existence of the localization. The existence of edge defects is distinguished by the location of eigenvalues of the time evolution operator.

SUMMARY: In this talk, we prove the essential self-adjointness of real principal type pseudo-differential operators in Eucliedan spaces under null non-trapping condition. For a proof, we employ microlocal propagation estimates in a interior region and a exterior region respectively. This is joint work with Shu Nakamura.

SUMMARY: We consider discrete Schrödinger operators on the \(2\)-dimensional hexagonal lattice, which is a tight-binding model of Hamiltonian on a graphene sheet. In this talk, we construct Isozaki–Kitada modifiers for a pair of the discrete Schrödinger operator without potential and that with a long-range potential.

SUMMARY: We consider the stableness for time-dependent propagator generated by the Klein–Gordon system with time dependent homogeneous electric fields, such a system can be described thorough the time-dependent non-selfadjoint operator.

SUMMARY: We study the resonances of a two-by-two semiclassical system of one dimensional Schrödinger operators, near an energy where the two potentials intersect transversally, one of them being bonding, and the other one anti-bonding. We obtain estimates on the location and on the widths of these resonances. Our method relies on the construction of fundamental solutions for the two scalar unperturbed operators, and on an iterative procedure in order to obtain solutions to the system.

SUMMARY: We propose a refined trace theorem which shows that the Fourier transform of weighted \(L^2({\bf R}^{n})\) functions can be regarded as \(L^2\) functions on each sphere \(|x|=r\) (\(n \geq 2\)) for a wide class of weighted functions. We discuss also about the Hölder continuity with respect to \(r\). As the application we consider uniform resolvent estimates of Dirac operators for the massive or massless case.

SUMMARY: We consider a generalized pair-interaction model \(H(\lambda )\) in quantum field theory with a coupling constant \(\lambda \in \mathbb {R}\). We talk about the spectrum of \(H(\lambda )\) for all \(\lambda \in \mathbb {R}\). In this talk, we introduce the operator \(H(\lambda )\) is diagonalized by a proper Bogoliubov transformation for some \(\lambda \in \mathbb {R}\). In particular, \(H(\lambda )\) has bound states for some \(\lambda \).

SUMMARY: The existence of the ground state of the so-called semi-relativistic Pauli–Fierz model is proven. Let \(A\) be a quantized radiation field and \(H_{{\rm f},m}\) the free field Hamiltonians which is the second quantization of \(\sqrt {|k|^2+m^2}\). The semi-relativistic Pauli–Fierz Hamiltonian is given by

\(H_{\rm SRPF}=\sqrt {(-i\nabla \otimes 1-A)^2+M^2} +V\otimes 1+1\otimes H_{{\rm f},m}\)

for \((m,M)\in [0,\infty )\times [0,\infty )\). We emphasize that our results include a singular case \((m,M)=(0,0)\).

SUMMARY: The class of power difference means, which is closely related to that of Stolarsky ones, includes typical operator means in the sense of Kubo–Ando. On the other hand, Wada pointed that the property ‘PMI’ implies the Ando–Hiai inequality, which becomes recently the remarkable one in the theory of multivariate operator means. In this talk, we restrict ourselves to representing operator monotone functions and discuss when the power difference means satisfy PMI or PMD.

SUMMARY: In this talk, we show some fundamental properties of the quantum Tsallis relative entropy of negative order based on the properties of the quasi geometric mean of positive semidefinite matrices. Moreover, we show matrix trace inequalities on the quantum Tsallis relative entropy of negative order, which includes the quasi geometric mean of positive definite matrices.

SUMMARY: For positive real numbers \(a\) and \(b\), the weighted power mean \(P_{t,q}(a,b)\) and the weighted Heron mean \(K_{t,q}(a,b)\) are defined as follows: For \(t \in [0,1]\) and \(q \in \mathbb {R}\), \(P_{t,q}(a,b)=\{ (1-t)a^{q}+tb^{q} \}^{\frac {1}{q}}\) and \(K_{t,q}(a,b)=(1-q)a^{1-t}b^{t}+q\{(1-t)a+tb\}\). These means generalize the arithmetic and the geometric ones.

In this talk, we get estimations of the weighted power mean by the Heron mean. In other words, we obtain the greatest value \(\alpha =\alpha (t,r)\) and the least value \(\beta =\beta (t,r)\) such that the double inequality \(K_{t,\alpha }(a,b)<P_{t,r}(a,b)<K_{t,\beta }(a,b)\) holds for \(t,r \in (0,1)\). We can also obtain operator inequalities for bounded linear operators on a Hilbert space.

SUMMARY: Let \(A\) and \(B\) be bounded positive invertible operators on a Hilbert space \(\mathcal {H}\), \(n\in \mathbb {N}\) and \(x\in \mathbb {R}\). We define the \(n\)-th version of the Tsallis relative operator entropy \(T_x(A|B)\) by \(T^{[1]}_x(A|B) \equiv T_x(A|B)\), \(T^{[n]}_x(A|B) \equiv \frac {T^{[n-1]}_x(A|B)-T^{[n-1]}_0(A|B)}{x} \ (x\neq 0)\) and \(T^{[n]}_0(A|B) \equiv \displaystyle \lim _{x\to 0}T^{[n]}_x(A|B)\) \((n\geq 2)\). In particular, \(T_0^{[n]}(A|B)\) is called the \(n\)-th relative operator entropy, and is denoted by \(S^{[n]}(A|B)\). In this talk, we show some properties of \(T^{[n]}_x(A|B)\) and \(S^{[n]}(A|B)\). Moreover, we try to introduce operator divergences defined by the differences between the \(n\)-th relative operator entropies, and show the properties and a relation between them.

SUMMARY: For \(C^1\)-function \(f(t)\) on an interval \(J\) and for \(t_0 \in J\), \(K_f(t, t_0):=\frac {f(t) - f(t_0)}{t - t_0}\). We will show that \(K_f(t, t_0)\) is strongly operator convex if and only if \(f\) is operator monotone.

SUMMARY: We give a limit formula for alpha determinants of a certain parametric deformation of normalized Laplacian matrices of complete graphs \(K_n\). We also calculate the normalized immanants of the same matrices, which allows us to obtain a limit formula for them. Further, we give a biclique-analog of the results above.

SUMMARY: We describe an invariant theory using the Cayley–Hamilton theorem of higher order (given by Y. Agaoka). We need this Cayley–Hamilton type theorem to describe the relations of generators of invariants as an ideal, though these relations can be described simply as an ideal with trace (as reported in the last MSJ meeting held in the spring 2018).

SUMMARY: Let \(G\) be a connected complex reductive Lie group with Lie algebra \(\mathfrak g\) and let \((\pi _\mu ,V_\mu )\) be a minuscule representation of \(G\). The space \(\mathcal P(\mathfrak g)\otimes V_\mu \) of \(V_\mu \)-valued polynomials on \(\mathfrak g\) is naturally a module of a commutative algebra \(\mathcal I_\mu \) containing \(\mathcal P(\mathfrak g)^G\) (A. A. Kirillov’s family algebra). In this talk, we define the space \(\mathcal H_\mu \) of \(V_\mu \)-valued harmonic polynomials and show the separation of variables formula “\(\mathcal P(\mathfrak g)\otimes V_\mu =\mathcal I_\mu \otimes \mathcal H_\mu \)” as a generalization of Kostant’s well-known result. We also discuss a natural system of generators for \(\mathcal H_\mu \), “restrictions” of \(\mathcal H(\mathfrak g)\) to various \(G\)-orbits in \(\mathfrak g\times V_\mu ^*\), and a generalization of the Hesselink–Peterson formula on graded multiplicities.

SUMMARY: We consider the realization of a minimal representation as the kernel of an intertwining differential operator from the space of smooth sections of an associated bundle over a flag manifold. It is shown that for the connected split real simple Lie groups of type \(D_n, E_6, E_7\) and \(E_8\), the space for an associated line bundle admits such realization if and only if the annihilator of an irreducible highest weight module for the corresponding Lie algebra is the Joseph ideal.

SUMMARY: Classical Hobson’s formula describes how a differential operator with constant coefficients acts on radial functions on an Euclidean space. We give an analogue of Hobson’s formula for Dunkl operators and its applications.

SUMMARY: We study discrete flow equivalence of two-sided topological Markov shifts by using extended Ruelle algebra, which is defined by a groupoid C*-algebra of an étale groupoid constructed from the Markov shift. We characterize flow equivalence of two-sided topological Markov shifts in terms of conjugacy of certain actions weighted by ceiling functions of two-dimensional torus on the stabilized extended Ruelle algebras for the two-sided topological Markov shifts.

SUMMARY: Minimal free topological dynamical systems are one of the major source to construct a natural simple C*-algebras (via the crossed product construction). In this talk, we clarify that for every pair of a non-torsion exact group and a compact space in a certain large class (e.g., the product of the Cantor set and topological closed manifold) there are continuously many minimal free actions which associate pairwise non-isomorphic groupoids. Moreover one can choose these actions to have a nice crossed product. Based on Appendix B of my paper arXiv:1702.04875.

SUMMARY: Let \(\mathcal {W}\) be the Razak–Jacelon algebra, which is a certain simple separable nuclear stably projectionless C\(^*\)-algebra having trivial \(K\)-groups and a unique tracial state and no unbounded traces. In this talk, we show that the central sequence C\(^*\)-algebra \(F(\mathcal {W})\) of \(\mathcal {W}\) is infinite.

SUMMARY: We study two subspace systems up to bounded isomorphism. For this purpose, it is crucially important to investigate operator ranges. It is related to a Hilbert representation of a Dynkin quiver. We introduce a bounded isomorphic invariant of two subspace systems constructed from graphs of Schatten class operators. We also point out that algebraic isomorphism and bounded isomorphism are different in these two subspace systems.

SUMMARY: We clarify the relation between the constructions by Bhat–Skeide and Muhly–Solel of minimal dilations of \(CP_0\)-semigroups.

SUMMARY: Since Wigner’s unitary-antiunitary theorem, there have been many researches on isometries between collections of projections in \(B(H)\). Recently, G. P. Gehér and P. Šemrl gave a complete description of surjective isometries (with respect to the operator norm) from the collection of all projections in \(B(H)\) onto itself. In this talk, we consider a further generalization of this result. Namely, we give a characterization of surjective isometries between projection lattices of two von Neumann algebras.

SUMMARY: We investigate the structure of the relative bicentralizer algebra \({\mathrm {B}}(N \subset M, \varphi )\) for inclusions of von Neumann algebras with normal expectation where \(N\) is a type III\(_1\) subfactor and \(\varphi \in N_*\) is a faithful state. We first construct a canonical flow \(\beta \) on the relative bicentralizer algebra and we show that the W\(^*\)-dynamical system \(({\mathrm {B}}(N \subset M, \varphi ),\beta ^\varphi )\) is independent of the choice of \(\varphi \) up to a canonical isomorphism. In the case when \(N=M\), we deduce new results on the structure of the automorphism group of \({\mathrm {B}}(M,\varphi )\) and we relate the period of the flow \(\beta ^\varphi \) to the tensorial absorption of Powers factors. For general irreducible inclusions \(N \subset M\), we relate the ergodicity of the flow \(\beta ^\varphi \) to the existence of irreducible hyperfinite subfactors in \(M\) that sit with normal expectation in \(N\). When the inclusion \(N \subset M\) is discrete, we prove a relative bicentralizer theorem and we use it to solve Kadison’s problem when \(N\) is amenable.

SUMMARY: We discuss the fullness of the crossed product von Neumann factor associated with a minimal action of a compact group on a factor.

SUMMARY: A closed ball with its center at the origin in a real Banach space \(X\) has the following property \((P)\): For every \(x\in X\), a positive-scalar multiple of \(x\) gives a nearest point in \(C\) to \(x\). In MSJ Spring Meeting 2018 at The University of Tokyo, we reported that a converse of this fact holds in the sense that a bounded closed convex set \(C\subset X\) with \(0\in \mathrm {Int}\,C\) possessing property \((P)\) is a closed ball with center \(0\), provided \(X\) is smooth and \(\dim \,X>1\)

In this talk we prove that the above result holds without the assumption of smoothness.

SUMMARY: We prove a geometric inequality for positive operators in the Banach algebra of all bounded linear operators on a Hilbert space. As an application we exhibit a several examples of a generalized gyrovector spaces (GGV) which consists of positive operators. We describe a surjective isometry for Thompson-like metric on these GGVs.

SUMMARY: We present Cauchy–Bunyakovsky–Schwarz type inequalities related to the M\(\rm \ddot o\)bius addition.