Hibert uzaylarında akretif formların operatörlerle temsili /
Yazar: BANU ÜNALMIŞ
Danışman: PROF.DR. MAHİR HASANOV
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü
Derece: Yüksek Lisans, 1995
Konu: Matematik = Mathematics
Dizin: Hilbert uzayları = Hilbert spaces

Özet:
ÖZET H Hubert uzayında Au = f (1) şeklinde bir denklem ele alınsın. Burada A : H -* H sınırsız operatör ve D(A) - H 'dır. (1) denkleminde A diferensiyel, integral, pseudodife- rensiyel vs. operatör olabilir. Bu tür denklemlerin farklı çözüm yol lan vardır. Bunlardan en önemlisi bilineer formların yardımıyla verilir. a[u, v] : H * H - > R1 bilineer form olsun. Eğer D{A) C D[a) ve u G D{A), v e D[a] için a[u, v] = (Au, v) şartı sağlanırsa o zaman A operatörü a[u,v] formu ile temsil edilir denir. a{u,v]=(f,v) (2) denklemini ele alalım. Eğer (1) denkleminin uq ? D {A) çözümü varsa o zaman uq elemanı a[ıto,u] = (f,v) denklemini de sağlar. Ama genelde tersi doğru değildir. Yani, Au = / denkleminin çözümü bulunmadığı halde (2) denkleminin ^o 6 D [a] gibi çözümü bulunur ve bu çözüme (1) denkleminin genelleştirilmiş çözümü denir. Bu nedenle (1) yerine (2) şeklindeki bilineer formlardan oluşmuş bir denklemi araştırmamız gerekiyor. Bu araştırmalar A operatörünün veya a[u, v] formunun hangi sınıf tan olduğuna bağlıdır. Eğer a[u, v] kuadratik formu pozitif definit olursa o zaman Friedrichs yöntemi kullanılır ve bu yöntem 1. bölümde ele alınmıştır. Akretif ve regüler akretif formlar ise 2. ve 3. bölümlerde incelenmiştir. iv

Abstract:
REPRESENTATION OF ACCRETIVE FORMS BY USING OPERATORS IN HILBERT SPACES SUMMARY This work is concerned with representing accretive and regularly accretive forms in a Hubert space by maximal accretive operators. In a Hilbert space H, let us consider an equation such as Au = /. (1) In this equation, A : H - » H is an unbounded operator and D{A) = H. In equation (1), A can be a differential, integral or pseudodifferential, etc. operator. As an example, if we define an operator A which satisfies the following conditions: ~ dx2 dy2 D(A) = {ue L2{ü)\ u e C2(Q),u\au = 0}, we can obtain the Dirichlet problem d2u d2u u\an = ° from the equation (1). These kinds of equations admits different solution methods. The most important of them is. given by bilinear forms. Let a[u, v] : H * H -* R1 is a bilinear form. II the condition a[u,v] = (Au,v) is satisfied for D (A) C D[a] and u ? D(A), v G D(A), then the operator A is said to be representable by the bilinear form a[u, v]. Consider the equation a[«,v] = (/,«). (2)If the equation (1) has a solution uq G D(A), then uq satisfies the equation a[uo,u] = (/, v). But in general, the reverse of it is not true. That is, even the equation Au = f has no solution, the equation (2) may have a solution uq G D[a}. This solution is called the generalized solution of the equation (1). The notion of the generalized solution is also defined in the theory of partial differential equations by some other means. As an example, let d2u d2u... dx* dy*~JV,yj (3) [u\s = 0 for (x, y)CttcR2 and S = dü. Multiply by the equation (3) by v G Cq°(CI) and integrate over the domain Q,, we obtain \dl?~ ~d1zjvdxdy = / f(x7VMx,y)dxdy. Using the Green formula in this integral equation, then the result will be / (ir + ^) ir = / /(X; yM*' y)dxdy- (4) The function u satisfying the equation (4) for all v G Cq°, is called the generalized solution of the equation (3). But defining the bilinear forms a[u,v] and (/,u) by du du \ dv., f ( du du\ dxi dyi ) dxi ' (M = / f(x,y)v(x,y)dxdy then the equation (4) can be written as a[u,v] = (/,v). So the notions of the generalized solutions defined by the equations (2) and (4) are the same. Since it furnishes us with a larger class of solutions, instead of (1) we are going to investigate the bilinear equation (2). These investigations are related to the class of the operator A or of the form a[u, v\. viIf the quadratic form a[«,u] is positive definite, then the so called method of Friedrichs will be used. This method is introduced in Chapter 1. In Section 1.1., the notions of the bilinear form and the quadratic form are defined and some examples are given. In Section 1.2., positive defi nite operators are defined. Positive definite operators are also symmetric operators. The symmetricity of a differential operator is determined by its statements as well as the boundary conditions accompanying it. In Examples 1.2.1. and 1.2. 2., how symmetric operators are made from dif ferent boundary conditions is indicated. In example 1.2.1., the operator A is defined by ? \ u(0) = u(l) = 0 is positive definite in H = £2(0, 1). In example 1.2.5., the operator B defined by d*u Bu = - -7-3 5 0 < a; < 00 viiis positive, but not positive definite in £2(0,00). The domain of B, D(B): are the functions which satisfy the following conditions: i)«eC^(o,oo), 2)u(0) = 0_, 3)There exists a constant au such that u(x) - 0 for all u G D{B), x > au. In Hubert space H, an energy space can be attached to the same as positive operator. This space is indicated by Ha- Ha is the domain of the bilinear form a[u,u]. That is, D[a] = H a- In Section 1.3., the energy spaces are studied. In general, the statement Ha C H, may not be true. But for the positive definite operators, H a C H is always be true (Theorem 1.3.1.). We choose the elements of Ha from the elements of H (Theorem 1.3.2.). According to this theorem, let A be a positive definite operator in Hubert space H. u G H is an element of energy space Ha if and only if there exists a«n6 D(A) such that |||«n-Um||| >0, ||«n-u|| >0 for n = 1, 2,... and n, m - > 00. If the equation Au = f has no solution with uq G D(A), then the operator A may be extended to a more extensive space to find the solution of the equation. This method, which is called Friedrichs extension, is introduced in Section 1.4. In Theorem 1.4.1., it is shown that after the extension of every positive definite operator to D(A), the equation Au = f has a solution for every / G H. Accretive and regularly accretive forms are given in Chapter 2 and Chapter 3. The operator Aj associated with a bilinear form J is defined as the operator with largest domain satisfying J[u,u] = (Aju,v) for all u G D(Aj),v G D(J), where D(A/) is the domain of Aj and D(J) is the domain of J. If J is accretive (Re J[u,u] > 0, u G D(J)), then Aj is also accretive (Re (AjU,u) > 0,u G D(Aj)). An accretive form J is defined to be representable if Aj is maximal accretive (i.e. has no proper accretive extension).. Our aim is to give conditions on J under which J is representable. Since a[u, v] is bounded for all u G H, a bounded accretive form on H is representable. So, from the Riesz theorem, the statement viiia[w,u] = (Au,v) is true for the bounded operator A. The first result of this kind for an unbounded form was derived by Friedrichs when he proved that the operator associated with a closed semibounded hermi- tian form is selfadjoint. This result was extended to a more general case by T. Kato who showed that a closed regularly accretive form is repre- sentable (see [5]). (Recall that an accretive form is regularly accretive if |Im J[w,u]| < 7 Re J[u,u] for all u 6 D{J) and some 7 > 0.) We consider accretive forms that are not necessarily regularly accre tive. Representation theorems for such forms are presented in Section 2.2. Then in Section 2.3. it is shown that the known result for regularly accretive forms is a corollary of these theorems. In Theorem 2.2.2., it is shown that the quadratic form J[«,u] is not accretive, i.e. Re J[u,u] $? Cr.Upto this point, J is accretive. So Re J[u,u] > 0. Indeed in this theorem, Re J[u7u] > a(u,u). In Section 2.4. we turn to the theory of partial differential equations. Throughout this section, Q, denotes an open subset of the m-dimensional euclidean space Rm, m > 1; H = £2(0); and fjk (j,k = l,---,m) are complex valued C1 functions defined on Q,. The real and imaginary parts of the matrix (fjk) are denoted by (gjk) and (hjk) respectively, i.e., 9jk = ^(fjk + fkj), hjk = T^ifjk ~ fkj)- Dj = d/dxj denotes differentiation in the generalized sense. So, if u is a locally integrable function, the statement "3Dyu "means "the distribu tion derivative Dju is a (locally integrable) function ". We investigate the form du dv '[«,«] = 1 x>^- n with the domain r ti,^ -,9«,.... f i v^. du dü i {uez'ms^-îorzn^dJlY,^^ 1, and fjk are C1 functions on H). Suppose that the values of ^ fjk(x)£j£k lie in a sector {z £ C|Re z > 0, |Im z| < 7Re z} of the complex plane C, for all £ G Cm IXand all x G O. Then J is known to be closed regularly accretive and hence representable (Theorem 2.4. 1.). We prove in Theorem 2.4.2. that J is representable even when this sector is allowed to rotate about zero as x varies in 0, so long as J remains accretive, and the rotation satisfies certain conditions. A simple example of a form that is not regularly accretive but which satisfies the hypotheses of this theorem is obtained when m - 1 and fl = (0, oo) on defining J by oo J[u,v]= / (1 + ixr sin.(xs)) - - dx 0 with r > 0, s > 0 and r + s < 1. There exist regularly accretive operators A in a Hilbert space H such that A1/2 and A*1/2 have different domains. Consequently, the domain of the closed bilinear form corresponding to A is different from the domain oîA1'2. In Chapter 3, we construct a regularly accretive operator A for which the domain of A*1/2 is different from the domain of A1/2. We remark that the domain of the closed bilinear form corresponding to such an operator A is also different from the domain of A1/2.