A Toeplitz-Like Operator with Rational Symbol Having Poles on the Unit Circle III: The Adjoint

This paper contains a further analysis of the Toeplitz-like operators $$T_\omega $$ T ω on $$H^p$$ H p with rational symbol $$\omega $$ ω having poles on the unit circle that were previously studied in Groenewald (Oper Theory Adv Appl 271:239–268, 2018; Oper Theory Adv Appl 272:133–154, 2019). Here the adjoint operator $$T_\omega ^*$$ T ω ∗ is described. In the case where $$p=2$$ p = 2 and $$\omega $$ ω has poles only on the unit circle $${\mathbb {T}}$$ T , a description is given for when $$T_\omega ^*$$ T ω ∗ is symmetric and when $$T_\omega ^*$$ T ω ∗ admits a selfadjoint extension. If in addition $$\omega $$ ω is proper, it is shown that $$T_\omega ^*$$ T ω ∗ coincides with the unbounded Toeplitz operator defined by Sarason (Integr Equ Oper Theory 61:281–298, 2008) and studied further by Rosenfeld (Classes of densely defined multiplication and Toeplitz operators with applications to extensions of RKHS’s, 2013; J Math Anal Appl 440:911–921, 2016).

Correct selfadjoint and positive extensions of nondensely defined minimal symmetric operators

LetA0be a closed, minimal symmetric operator from a Hilbert spaceℍintoℍwith domain not dense inℍ. LetA^also be a correct selfadjoint extension ofA0. The purpose of this paper is (1) to characterize, with the help ofA^, all the correct selfadjoint extensionsBofA0with domain equal toD(A^), (2) to give the solution of their corresponding problems, (3) to find sufficient conditions forBto be positive (definite) whenA^is positive (definite).

On the domain of selfadjoint extension of the product of Sturm-Liouville differential operators

The second-order symmetric Sturm-Liouville differential expressionsτ1,τ2,…,τnwith real coefficients are considered on the intervalI=(a,b),−∞≤a<b≤∞. It is shown that the characterization of singular selfadjoint boundary conditions involves the sesquilinear form associated with the product of Sturm-Liouville differential expressions and elements of the maximal domain of the product operators, and it is an exact parallel of the regular case. This characterization is an extension of those obtained by Everitt and Zettl (1977), Hinton, Krall, and Shaw (1987), Ibrahim (1999), Krall and Zettl (1988), Lee (1975/1976), and Naimark (1968).

QUANTUM 'ax + b' GROUP

'ax + b' is the group of affine transformations of the real line R. In quantum version ab = q2ba, where q2 = e-i ℏ is a number of modulus 1. The main problem of constructing quantum deformation of this group on the C*-level consists in non-selfadjointness of Δ(b) = a ⊗ b + b ⊗ I. This problem is overcome by introducing (in addition to a and b) a new generator β commuting with a and anticommuting with b. β (or more precisely β ⊗ β) is used to select a suitable selfadjoint extension of a ⊗ b + b ⊗ I. Furthermore we have to assume that [Formula: see text], where k = 0,1,2, ·. In this case, q is a root of 1. To construct the group, we write an explicit formula for the Kac–Takesaki operator W. It is shown that W is a manageable multiplicative unitary in the sense of [3,19]. Then using the general theory we construct a C*-algebra A and a comultiplication Δ ∈ Mor (A,A ⊗ A). A should be interpreted as the algebra of all continuous functions vanishing at infinity on quantum 'ax + b'-group. The group structure is encoded by Δ. The existence of coinverse also follows from the general theory [19].

QUANTUM EXPONENTIAL FUNCTION

A special function playing an essential role in the construction of quantum "ax+b"-group is introduced and investigated. The function is denoted by Fℏ(r,ϱ), where ℏ is a constant such that the deformation parameter q2=e-iℏ. The first variable r runs over non-zero real numbers; the range of the second one depends on the sign of r: ϱ=0 for r>0 and ϱ=±1 for r<0. After the holomorphic continuation the function satisfies the functional equation [Formula: see text] The name "exponential function" is justified by the formula: [Formula: see text] where R, S are selfadjoint operators satisfying certain commutation relations and [R+S] is a selfadjoint extension of the sum R+S determined by operators ρ and σ appearing in the formula. This formula will be used in a forthcoming paper to construct a unitary operator W satisfying the pentagonal equation of Baaj and Skandalis.

Necessary and Sufficient Conditions for the Discreteness of the Spectrum of Certain Singular Differential Operators

1. Introduction. Let P(x) be an m × m matrix-valued function that is continuous, real, symmetric, and positive definite for all x in an interval J , which will be further specified. Let w(x) be a positive and continuous weight function and define the formally self adjoint operator l bywhere y(x) is assumed to be an m-dimensional vector-valued function. The operator l generates a minimal closed symmetric operator L0 in the Hilbert space ℒm2(J; w) of all complex, m-dimensional vector-valued functions y on J satisfyingwith inner productwhere . All selfadjoint extensions of L0 have the same essential spectrum ([5] or [19]). As a consequence, the discreteness of the spectrum S(L) of one selfadjoint extension L will imply that the spectrum of every selfadjoint extension is entirely discrete.