| Yan Shaozong.[J].数学年刊A辑,1980,1(3-4):485~499 |
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| ON PGLAR PRODUCT OPERATOR |
| Received:November 27, 1979 Revised:January 24, 1980 |
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| Let H be a Hilbert space, and let A be a linear bounded operator on H. For
\(\lambda \in \rho (A)\), the \({U_\lambda } = {(A - \lambda )^{ * - 1}}(A - \lambda )\) is called polar.Produot operator.
In this paper, we discuss the properties of \({U_\lambda }\) and the relation between \({U_\lambda }\) and A. We obtain tbe following results.
Definition. Let B be a linear bounded operator on H, suppose \(0 \in \rho (B)\). For every
\(x,y \in H\), we definite \([x,y] = (Bx,y)(H,B)\)(or (H, [·,·]) is called a non-
degenerate bilinear space (it is obvious that if B=B*，then (H，B)is a space with an
indefinite metric; and that if B>0, then (H，B) is a Hilbert Space. If an operator
U(A) satisfies
\[[Ux,Uy] = [x,y]([Ax,y] = [x,Ay]),x,y \in H\]
then the operator U(A) is called a wvitary (self adjoint) on (H,B).
Theorem I . Suppose A is a linear bounded operator on H,
(1) If \(0 \in \rho (A)\), then \(U = {A^{ * - 1}}A\) is a unitary operator on (H，A) or (H, A*), and \(\sigma (U) = \frac{1}{{\sigma (U)}}\).
(2) If there is a complex number \(\alpha \), such that \({\mathop{\rm Im}\nolimits} A \ge \alpha > 0\) then a)\(0 \in \rho (A)\), and
the operator \(U = {A^{ * - 1}}A\) is a unitary on Hilbert space \((H,{\mathop{\rm Im}\nolimits} A) and 1 \in \rho (U)\)；b) there exist two Hilbert spaces \((H,{v_1}),(H,{v_2})\), such that A, A* are all the unitary operator
from (H,v1) onto (H,v2), and there are two spectral measures \(\{ E_\lambda ^i,\lambda \in [\alpha ,\chi ] \subset (0,2\pi )\} ,i = 1,2,\), such that \(AE_\Delta ^1H \subset E_\Delta ^2H,{A^ * }E_\Delta ^1H \subset E_\Delta ^2H\) for any \(\Delta = (\lambda ,u] \subset (0,2\pi ]\).
(3) If \(0 \in \rho (A) \cap \rho ({\mathop{\rm Im}\nolimits} A)\) then the operator \(U = {A^{ * - 1}}A\) is a unitary on \((H,{\mathop{\rm Im}\nolimits} A)\).
with an indefinite met He, and \(1 \in \rho (U)\).
(4) For any complex number \(\lambda = r{e^{i\theta }},\left| \lambda \right| > \left\| A \right\|\), then \({U_\lambda }\) must be a unitary operator on the Hilbert space \(\left( {H,{\mathop{\rm Im}\nolimits} (\frac{1}{{i\lambda }}A + iI)} \right),and - {e^{i2\theta }} \in \rho ({U_\lambda })\)
Theorem 2. (1) A is a normal operator iff there exists a complex number \(\lambda \),
\(\lambda \in \rho (A)\), such that \(\frac{{\partial {U_\lambda }}}{{\partial \lambda }}{U_\lambda } = {U_\lambda }\frac{{\partial {U_\lambda }}}{{\partial \lambda }}\),where \(\frac{\partial }{{\partial \lambda }}\) is the directional derivative.
(2) If there exists a complex number \(\alpha ,{\mathop{\rm Im}\nolimits} A \ge \alpha > 0\) then A is a normal operator
iff \(U = {A^{ * - 1}}A\) is also.
(3) If A is a hyperiwrmal or a subnormal, then for every \(\lambda \in \rho (A),\sigma ({U_\lambda })\) lies on the circle.
3, 4期
关于极?积算子炉一U
499
Theorem 3? Let A be a linear bounded operator on Hm Suppose 0￡p(J.)?
(1) If U=А*~гА is a unitary operator in a certain non-degenerate bilinear space
(H} 1 J |Л|>1 入 J Ш=1
solmble iff (l)cr(ü) =-=i=-, (2) the operotor Ui= f XdE\ is unitarity equivalent to
cr(C7) J |л|>1
the operator ül_1 == ШЕ1, 、
J 1Л|>1
If the conditions (1)，(2) are satified, then we have
(3) the subspace JJi?^2 of H reduces any solution of the equation A*"1 A — U, and
1
AI (Ягея*)1 = ^Uo where Ъ is any in vertible self-adjoint on (i?i?J?2) L, and bv Uo
(Uo = Jiai i ЫЕ1 )}⑷41Я1фя, = , Ла = A1ü2, if the operator V is to
realize TJt and Ut^unitary equivalence^ then Ai = VSf where S is any invertible on Нг
and SvUi. "
Corollary. Swpp)se operator U is a normal on a certain Hilbert space(Hy v) (^the
U is similar to a certain normal operator on IT), if U satisfies the conditions (1)、(2) of
Theorem^ on ?S，v) ? Then the general form of the solution of А*~гА = U is A = vA'y
where A' is same as A in Theorem^.
Theorem 5. Let U be a linear bounded operator on H. Suppose O?p(J.)and p(ü)
is a simply connected region, then the equation А*~гЛ = U is a solvable iff there
exists a certain space {H，v) with a indefinite metric, such that U is a unitary operator
on (Hf 丨）.
If is a unitary operator on ?H，v)，then there exists a particular solution of
I X+$
А*~гА = U: Ar = 2e v [ (?7 —X)_1+X], where eie ? p (JJ), and the general form of the
solution is Аж АУ, wliere V is any in vertible self-adjoint on (H, -u)and VvU, |
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