Numerical range and accretivity of operator products

On the solvability of fourth-order boundary value problems with accretive operators

“…This means that there exist α ≥ 0 and 0 ≤ β < 1 such that In general, statement of Theorem 3.4 is not true if we only assume that B is accretive. In fact B 2 + C not need to be m-accretive, because B 2 fails to be accretive (with the same vertex) even in the case of an accretive matrix B with numerical range contained in a sector of angle lesser than π/4, see Example 1.2 in [11].…”

“…Such results are interesting, firstly because they provide an existence, uniqueness and L p -maximal regularity of the solution for the fourth order abstract boundary value problem (1.1) under some necessary and sufficient conditions on the data. Moreover, they are of interest regarded as an application of the perturbation theory of m-accretive operators, [10], [11], [13], [20], [21], [22], [32]. This allows us to find various sufficient conditions on the accretive operators B and C, under which our results remains valid.…”

On the joint antieigenvalues of operators in normal subalgebras

“…The quantity µ 1 (T ) is also denoted by cos T and is called the cosine of T . The quantity µ 1 (T ) has important applications in the study of the numerical range of operators and numerical methods for optimization (see [3,7,8]). It is proved in Gustafson [5] that for an accretive operator T we have sin T = √ 1 − cos 2 T , where sin T is defined by sin T = inf >0 T − I .…”

Computation of Antieigenvalues of Bounded Linear Operators Via Centre of Mass

“…The first antieigenvalue can be interpreted as the cosine of the largest angle ( real ) through which any vector can be rotated by the action of T . The concept of antieigenvalues is studied by Gustafson [8][9][10][11][12], Gustafson and Rao [13][14], Gustafson and Seddighin [15], Das et al [2] ,Paul [22], Paul and Das [23].…”