On the base of boundary-value, spectral and initial-boundary value problems studied earlier for case single domain, we consider corresponding generated by a sesquilinear form two domains. Arising operator pencils with coefficients acting in Hilbert space depending on parameters are detail. In perturbed unperturbed cases, situations where one is other fixed. this paper, use superposition princip...

Positive operator measures (with values in the space of bounded operators on a Hilbert space) and their generalizations, mainly positive sesquilinear form measures, are considered with the aim of providing a framework for their generalized eigenvalue type expansions. Though there are formal similarities with earlier approaches to special cases of the problem, the paper differs e.g. from standar...

Over a field or skew field F with an involution a 7→ ã (possibly the identity involution), each singular square matrix A is *congruent to a direct sum S∗AS = B ⊕ Jn1 ⊕ · · · ⊕ Jnp , 1 ≤ n1 ≤ · · · ≤ np, in which S is nonsingular and S∗ = S̃ ; B is nonsingular and is determined by A up to *congruence; and the ni-by-ni singular Jordan blocks Jni and their multiplicities are uniquely determined by ...

We consider the problem of maximal regularity for semilinear non-autonomous fractional equations $$\begin{aligned} B^\alpha u(t)+A(t)u(t)=F(t,u),\, t \text {-a.e}. \end{aligned}$$ Here $$B^\alpha $$ denotes Riemann–Liouville derivative order $$\alpha \in (0,1)$$ w.r.t. time and time- dependent operators A(t) are associated with (time dependent) sesquilinear forms on a Hilbert space $${\mathcal ...

We show that the Lawrence–Krammer representation is unitary. We explicitly present the non-singular matrix representing the sesquilinear pairing invariant under the action. We show that reversing the orientation of a braid is equivalent to the transposition of its Lawrence–Krammer matrix followed by a certain conjugation. As corollaries it is shown that the characteristic polynomial of the Lawr...

Let (A,Ao) be a topological quasi *-algebra, which means in particular that Ao is a topological *-algebra, dense in A. Let πo be a *-representation of Ao in some pre-Hilbert space D ⊂ H. Then we present several ways of extending πo, by closure, to some larger quasi *-algebra contained in A, either by Hilbert space operators, or by sesquilinear forms on D. Explicit examples are discussed, both a...

encountered in the course of studying weak formulations of differential equations. Unlike the differential expressions, the theory behind the sesquilinear forms (1.2) is not yet fully developed. The most general treatment we have so far is for the case when such forms are semibounded or sectorial [10]. The classical Lax-Milgram theorem which is widely used in treatments involving the bilinear f...

The aim of these two papers (I and II) is to try to give fundamental concepts of quantum kinematics to q-deformed quantum spaces. Paper I introduces the relevant mathematical concepts. A short review of the basic ideas of q-deformed analysis is given. These considerations are continued by introducing q-deformed analogs of Fourier transformations and delta functions. Their properties are discuss...