TY - JOUR

T1 - Control subspaces of minimal dimension and root vectors

AU - Nikol'skil, N. K.

AU - Vasjunin, V. I.

PY - 1983/12/1

Y1 - 1983/12/1

N2 - We investigate the following characteristic of a linear operator A in a Banach space X: disc[Figure not available: see fulltext.] {inf(dim R′:R′⊂R,R′εCyc A) :RεCyc A}, where Cyc A={R:R is a subspace of X, dim R<∞, span (AnR: :n≥0)=X}. The value disc A is equal to the dimension of a cyclic subspace that can be chosen in an arbitrary cyclic finite dimensional subspace. If we consider a dynamical system {Mathematical expression} with the controllability property, disc A shows to what extent the dimension of the input subspace of control can be diminished without loss of controllability. In this paper we investigate when easy inequality disc A≥(the multiplicity of A) turn into the equality. Some estimates from below of disc A (of the type disc A≥sup dim Ker(A-λI)) are found for some classes of operators e.q. for compact operators, for Toeplitz operators with antianalytic symbols, for strictly lower triangular operators and some other classes.

AB - We investigate the following characteristic of a linear operator A in a Banach space X: disc[Figure not available: see fulltext.] {inf(dim R′:R′⊂R,R′εCyc A) :RεCyc A}, where Cyc A={R:R is a subspace of X, dim R<∞, span (AnR: :n≥0)=X}. The value disc A is equal to the dimension of a cyclic subspace that can be chosen in an arbitrary cyclic finite dimensional subspace. If we consider a dynamical system {Mathematical expression} with the controllability property, disc A shows to what extent the dimension of the input subspace of control can be diminished without loss of controllability. In this paper we investigate when easy inequality disc A≥(the multiplicity of A) turn into the equality. Some estimates from below of disc A (of the type disc A≥sup dim Ker(A-λI)) are found for some classes of operators e.q. for compact operators, for Toeplitz operators with antianalytic symbols, for strictly lower triangular operators and some other classes.

UR - http://www.scopus.com/inward/record.url?scp=34250150102&partnerID=8YFLogxK

U2 - 10.1007/BF01691899

DO - 10.1007/BF01691899

M3 - Article

AN - SCOPUS:34250150102

VL - 6

SP - 274

EP - 311

JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

SN - 0378-620X

IS - 1

ER -