none

關鍵字: 次算子, 次正規算子

In this thesis, we study the extension properties of a bounded linear
transformation from a subspace of a Hilbert space into the whole space(e.g., which has a normal
extension). Given an $n imes n$ normal matrix $A$ and a $k imes n$ matrix $B$, $kleq n$, we obtain
some sufficient conditions of subnormality for the submatrix(column matrix)
$left[ egin{array}{c}
A
Bend{array}
ight]$
by means of the geometric behavior of $A$ and $B$. If, in particular, $B$ is
of rank one, we show that these sufficient conditions are also necessary for subnormality of
$left[ egin{array}{c}
A
Bend{array}
ight]$.
In order to prove these results, we establish the key lemma which says that $XX^*=B^*B$ if and only
if $X^*=VB$ for some $k imes k$ unitary matrix $V$.

The submatrix $left[ egin{array}{c}
A
Bend{array}
ight]$
can be regarded as a ``partial matrix" defined in [13]. Therefore, the extension
problem about suboperators is also the matrix completion problem which is just the question as to
whether a partial matrix has a completion in a certain class or with a certain property of interest.
In the history of the matrix completion problem(or extension problem), the most part of related bibliographies
have no concern with ``normal completion" or ``normal extension" which may be the hardest and most important
questions about suboperators. There are many interesting questions in this subject to be explored.

Keywords: subnormal, submatrix, suboperator

[1] J. B. Conway, A Course in Functional Analysis, Second ed., Springer-Verlag, New York, 1990.
[2] J. B. Conway, The Theory of Subnormal Operators, MATHEMATICAL Surveys and Monographs, vol. 36, American Mathematical Society, USA, 1991.
[3] H. Dym, I. Gohberg, Extensions of band matrices with band inverses, Linear Algebra
Appl. 36(1981), 1-24.
[4] R. Ellis, D. Lay, Rank Preserving Extensions of Band Matrices, Linear and Multilinear
Algebra, 26(1990), no. 3, 147-179.
[5] S. H. Friedberg and A. J. Insel, Hyponormal $2 imes{2}$ matrices are
subnormal, Linear Algebra Appl. 175:31-38(1992).
[6] B. Fuglede, A commutativity theorem for normal operators, Proc. Nat. Acad. Sci.,vol.36(1950), 35-40.
[7] R. Grone, C. R. Johnson, E. M. Sa, H. Wolkowicz, Normal Matrices, Linear Algebra Appl. (1987)213-225.
[8] R. Grone, C. R. Johnson, E. S${
m acute{a}}$ and H. Wolkowicz, Positive definite
completions of partial Hermitian matrices, Linear Algebra Appl. 58(1984), 109-124.
[9] M. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.
[10] P. R. Halmos, Subnormal suboperator and the subdiscrete topology, Internat. Ser. Numer.
Math. 65, Birkh${
m ddot{a}}$user , Basel, 1984, pp. 49-65.
[11] P. R. Halmos, A Hilbert Space Problem Book, Second ed., Springer-Verlag, New York, 1982.
[12] K. Hoffman, R. Kunze, Linear Algebra, Second ed., P{scriptsize RENTICE}-H{scriptsize ALL}, I{scriptsize NC}, Englewood Cliffs, New Jersey, 1971.
[13] C. R. Johnson, Matrix completion problems: A Survey, Proc. Symposia Appl. Math.1990,171-198.
[14] C. R. Johnson, M. Lundquist, Numerical Ranges for Partial Matrices, Topic in matrix and operater theory(Rotterdam 1989), 283-291, Oper. Theory Adv.Appl.,50,Birkh${
mddot{a}}$user, Basel, 1991.
[15] C. S. Kubrusly, An Introduction to Models and Decompositions in Operator Theory,
Birkh${
m ddot{a}}$user, 1997.
[16] M. Martin, M. Putinar, Lectures on Hyponormal Operators, Birkh${
m ddot{a}}$user, 1989.

[17] N. G. Makarov, Perturbations of normal operators and stability of the continuous
spectrum, English transl. Math. URRS-Izv. 29 (1987), 535-558.
[18] N. K. Nikolskii, On perturbations of the spectrum of unitary operators, Mat
Zametki 5, 341-349; English transl. Math. Notes 5 (1969), 207-211.
[19] B. P. Rynne, M. A. Youngson, Linear Functional Analysis, Springer-Verlag London Berlin Heidelberg, 2000.
[20] D. Serre, Matrices: Theory and Applications, Springer-Verlag, New York, 2002.
[21] T. Yoshino, Introduction to Operator Theory, Longman Scientific & Technical, England, 1993.