Spectral theory of compact linear operators and applications

Project Abstract

Spectral theory of compact linear operators is a fundamental area of functional analysis that plays a crucial role in various branches of mathematics, physics, and engineering. It provides a powerful framework for understanding the behavior of linear operators on infinite-dimensional spaces. In this research project, we investigate the spectral properties of compact linear operators on Hilbert spaces and explore their applications in different fields. One of the key aspects of spectral theory is the decomposition of a compact operator into its spectral components, which allows us to study the operator's behavior in terms of its eigenvalues and eigenvectors. We analyze the spectrum of compact operators, including the essential spectrum, point spectrum, and residual spectrum, and study the convergence of spectral approximations for these operators. Moreover, we examine the spectral mapping theorem for compact operators, which establishes a connection between the spectra of an operator and its powers, exponential, and functional calculus. This theorem provides a powerful tool for analyzing the spectral properties of operators and understanding their behavior under various transformations. Furthermore, we investigate the Fredholm theory for compact operators, which characterizes the solvability of linear equations involving compact operators. This theory plays a crucial role in understanding the invertibility and range of compact operators and has important applications in areas such as integral equations and differential equations. In addition to theoretical aspects, we explore various applications of spectral theory in different fields. In functional analysis, spectral theory is used to study the properties of integral operators, differential operators, and other linear operators on function spaces. In quantum mechanics, spectral theory is essential for understanding the behavior of quantum mechanical systems and analyzing the spectra of quantum observables. Moreover, spectral theory finds applications in signal processing, image processing, and machine learning, where it is used for analyzing and processing signals and images using linear operators. The study of compact operators and their spectral properties provides valuable insights into the behavior of these operators in practical applications and enables the development of efficient algorithms for signal and image processing tasks. Overall, the spectral theory of compact linear operators is a rich and diverse area of research with wide-ranging applications in mathematics, physics, and engineering. By studying the spectral properties of compact operators and their applications, we gain a deeper understanding of linear operators on infinite-dimensional spaces and develop powerful tools for solving various mathematical problems.

Project Overview

<p> </p><p>FOUNDATION<br>1.1 Basic notions and results from Functional<br>Analysis<br>The purpose of this section is to refresh our minds on some basic fundamentals<br>required to facilitate a smooth understanding of the study of compact linear<br>operators and its applications.<br>Denition 1.1.1. A non-negative function jj jj on a vector space X is called<br>a norm on X if and only if<br>i) jjxjj 0 for every x 2 X (Positivity).<br>ii) jjxjj = 0 if and only if x = 0 (Nondegeneracy).<br>iii) jj(x)jj = jjjjxjj for every x 2 X (Homogeneity).<br>iv) jjx + yjj jjxjj + jjyjj for every x; y 2 X (subadditivity).<br>A vector space X with a norm jj jj is denoted by (X; jj jj) and is called a<br>normed linear space (or just a normed space).<br>A sequence (xn)n2N of elements in a normed linear space X is called Cauchy<br>if 8 &gt; 0; 9N 2 N, such that for m; n 2 N, jjxm ô€€€ xnjj , m; n N<br>A Banach space is a normed linear space (X; jj jj) that is complete in the<br>canonical metric dened by (x; y) = jjx ô€€€ yjj for x; y 2 X i.e every Cauchy<br>sequence in X for the metric converges to some point in X.<br>1<br>Remark. Every normed linear space has a completion [3]<br>Denition 1.1.2. Let X and Y be K-linear spaces (K = R or K = C). A<br>map T : X ô€€€! Y is called linear if<br>T(f + g) = T(f) + T(g) for all f; g 2 X; and all ; 2 K:<br>More generally, we can consider linear maps dened on a sub-space D(T)<br>of X and with values in Y . The Subspace D(T) X is called the domain of<br>T. We denote the image (or Range) of T by R(T) and is dened by<br>R(T) :=</p><p>y 2 Y : y = Tx for some x 2 D(T)</p><p>:<br>We dene the Kernel (or Null space) of T denoted by N(T) to be the subspace<br>of X dened by<br>N(T) := fx 2 D(T) : Tx = 0g<br>T is said to be injective (or one to one ) if N(T) = f0g.<br>T is said to be surjective (or onto) if R(T) = Y .<br>Denition 1.1.3. A mapping T : X ô€€€! Y is called a continuous linear<br>operator, if T is linear and is continuous at each point in a 2 X, that is<br>lim<br>x!a<br>Tx = Ta for all a 2 X :<br>The space of continuous linear operators from a Banach space X into a<br>Banach Space Y is denoted by B(X; Y ).<br>Proposition 1.1.4. [2]<br>Let X, Y be two normed spaces and BX be the closed unit ball of X. Let<br>T : X ô€€€! Y be a linear map. Then the following are equivalent :<br>i) T is a bounded linear operator, i.e there exists a constant M &gt; 0 such that<br>jjTxjjY MjjxjjX for all x 2 X.<br>ii) T is continuous.<br>iii) T is continuous at the origin (in the sense that if fxng is a sequence in<br>X. such that xn ! 0 as n ! 0 , then Txn ! 0 in Y as n ! 1).<br>iv) T is Lipschitz.<br>v) T(BX) is bounded (i.e there exists a constant C &gt; 0 such that jjTxjj C<br>for all x 2 BX).<br>2<br>Furthermore, B(X; Y ) becomes naturally endowed with the operator norm,<br>jjTjj := sup<br>x2X;x6=0<br>jjTxjjY<br>jjxjjX<br>= sup<br>jjxjjX1<br>jjTxjjY ;<br>and if Y is a Banach space then so is B(X; Y ).<br>We also recall that every linear operator of a nite dimensional space is bounded.<br>Proposition 1.1.5. Every bounded linear map between normed linear space<br>has a unique extension between their completions [4].<br>Theorem 1.1.6. Given a continuous complex function G on [a; b] [a; b],<br>let T be dened on X = C[a; b] at each f by<br>(Tf)(x) =<br>Z b<br>a<br>G(x; y)f(y)dy for all x 2 [a; b]:<br>T is called an integral operator with the kernel function G. Then T 2 B(X; Y )<br>and<br>jjTjj = max<br>axb<br>Z b<br>a<br>jG(x; y)jdy<br>Proof.<br>Firstly, we show that T is well dened, i.e., for every f 2 C[a; b], Tf 2 C[a; b].<br>Let x1; x2 2 [a; b], then<br>jTf(x1) ô€€€ Tf(x2)j<br>Z b<br>a<br>jG(x1; t) ô€€€ G(x2; t)jjf(t)jdt<br>max<br>t2[a;b]<br>jG(x1; t) ô€€€ G(x2; t)jjjfjj1(b ô€€€ a)<br>Since G is continuous on the compact set C([a; b] [a; b]) by assumption, it<br>follows that G is uniformly continuous and so we deduce from the above inequality<br>that Tf is uniformly continuous. Thus Tf 2 C[a; b].<br>The linearity of T follows from the linearity of the integral.<br>Now we investigate the boundedness of T. We have,<br>j(Tf)(x)j<br>Z b<br>a<br>jG(x; y)jdy (1.1.1)<br>and so,<br>jjTfjj1 = max<br>axb<br>j(Tf)(x)j max<br>axb<br>Z b<br>a<br>jG(x; y)jdyjjfjj1<br>showing that T is bounded and<br>kTk max<br>axb<br>Z b<br>a<br>jG(x; y)jdy :<br>3<br>Hence<br>kTk M; where M :=<br>Z b<br>a<br>jG(x; y)jdy:<br>Now we dene,<br>S(x) :=<br>Z b<br>a<br>jG(x; y)jdy:<br>S is continuous on [a; b] and therefore it attains its maximum at some x0 2<br>[a; b]:<br>Let g(y) :=<br>8&lt;<br>:<br>jG(x0;y)j<br>G(x0;y) if G(x0; y) 6= 0<br>0 otherwise<br>Then the function g is bounded and Lebesgue measurable and so belongs to<br>L1([a; b]). It follows from the fact that C[a; b] is dense in L1([a; b]) that there<br>exists a sequence (gn)n of elements of C[a; b] such that kgnk = max<br>y2[0;1]<br>jgn(y)j 1<br>and (gn) converges in L1([a; b]) to g. Hence we have ,<br>kTk = sup<br>jjfjj1<br>kTfk kTgnk = max<br>axb<br>j(Tgn)xj (Tgn)(x0)<br>) kTk (Tgn)(x0) =<br>Z b<br>a<br>K(x0; y)gn(y)dy ô€€€!<br>Z b<br>a<br>K(x0; y)g(y)dy<br>therefore, kTk<br>Z b<br>a<br>K(x0; y)g(y)dy =<br>Z b<br>a<br>jK(x0; y)jdy = M<br>It follows that jjTjj M: Hence,<br>kTk = M<br>Corollary 1.1.7. Given a continuous complex valued function G on<br>[0; 1] [0; 1], let T be dened on X = C[0; 1] by<br>(Tf)(x) =<br>Z 1<br>0<br>G(x; y)f(y)dy ; 8f 2 C[a; b]:<br>Then T is a bounded linear map with<br>jjTjj = max<br>0x1<br>Z 1<br>0<br>jG(x; y)jdy :<br>Denition 1.1.8. A map T dened from a Banach space X into a Banach<br>space Y is called closed if its graph<br>G(T) = f(x; y) : x 2 D(T)g<br>is closed in X Y . In other words, T is closed if whenever xn ô€€€! x and<br>Txn ô€€€! y, we have x 2 D(T) and Tx = y<br>4<br>Remark. Every T 2 B(X; Y ) is closed<br>Theorem 1.1.9. Closed Graph Theorem<br>A closed linear map which maps a Banach space into a Banach space is con-<br>tinuous<br>Denition 1.1.10. A map T 2 B(X; Y ) is invertible if there is a bounded<br>linear map Tô€€€1 2 B(Y;X) such that Tô€€€1T = IX (the identity of X) and<br>TTô€€€1 = IY (the identity operator of Y )<br>Corollary 1.1.11. Every continuous bijection between Banach spaces has a<br>continuous inverse.<br>Proof.<br>Let T 2 B(X; Y ). Then G(T) = f(x; Tx) : x 2 Xg is closed in X Y ,<br>G(Tô€€€1) = f(Tx; x) : x 2 Xg is closed in Y X<br>Tô€€€1 is a closed linear operator mapping Y into X.Therefore Tô€€€1 is bounded<br>by the closed graph theorem.<br>Denition 1.1.12. Let A be a subset of a Banach space X. A is said to<br>be precompact if any sequence of A has a Cauchy subsequence. A is totally<br>bounded if for every &gt; 0, there exists a nite cover for A of open balls of<br>same radius<br>Denition 1.1.13. A is said to be compact if any sequence of A has a subse-<br>quence that converges to some point of A.<br>Remark. A set is said to be precompact in a Banach X if and only if its closure<br>is compact in X.<br>Proposition 1.1.14. Let A be a subset of a Banach space X. A is said to be<br>precompact if and only if it is totally bounded.<br>Proof.<br>Assume that A is a precompact subset of X, we show that A is totally<br>bounded.<br>Let &gt; 0 and x 2 A<br>x 2 A ) B(x; )</p><p>A 6= ;<br>) 9 ax 2 A : ax 2 B(x; )<br>) 9 ax 2 A : d(ax; x) &lt;<br>) 9 ax 2 A : x 2 B(ax; )<br>[<br>a2A<br>B(a; )<br>since x was arbitrarily chosen, we have that A<br>[<br>a2A<br>B(a; ), using the compactness<br>of A; 9 a1; a2:::an 2 A such that A<br>[n<br>i=1<br>B(ai; ) ) A<br>[n<br>i=1<br>B(ai; )<br>5<br>since A A.Hence A is totally bounded.<br>Conversely, suppose A is totally bounded, we show that A is precompact<br>(i.e A is compact).<br>Let fangn A, we show that fangn has a Cauchy subsequence<br>Case I. fan; n 1g is nite<br>Then there exists n1; ::; np elements of N such that<br>fan : n 1g = fan1 ; ::; anpg:<br>Dene for each i 2 f1; ::pg, Ei := fn 2 N : an = anig.<br>So we have N =<br>Sp<br>i=1 Ei.<br>Since N is innite, then one of the sets Ei is innite. Choose i0 2 f1:; ; :pg such<br>that Ei0 is innite.<br>Since Ei0 N, then it has a minimum element. Let m1 := minEi0<br>For k 1, mk+1 := minfEi0 n fm1;m2:::mkgg<br>Clearly, mk &lt; mk+1, 8k 2 N, also amk = ani0 since mk 2 Ei0<br>famkgk1 is a constant subsequence of fang which is convergent.<br>Case II. fan; n 1g is innite.<br>For = 1<br>22 &gt; 0; 9 x1; x2:::xm such that A<br>m[<br>i=1<br>B(xi;<br>1<br>22 ) since A is totally<br>bounded.<br>This implies that A<br>m[<br>i=1<br>B<br>(xi;<br>1<br>22 ) since B(xi; 1<br>22 ) B<br>(xi; 1<br>22 ).<br>So A<br>m[<br>i=1<br>B<br>(xi;<br>1<br>22 ) since<br>m[<br>i=1<br>B<br>(xi;<br>1<br>22 ) is closed.<br>It follows that A<br>m[<br>i=1<br>B(xi;<br>1<br>2<br>) which implies that<br>fangn<br>m[<br>i=1<br>B(xi;<br>1<br>2<br>), since fangn A<br>Therefore, 9 i0 2 f1; 2:::mg such that B(xio ; 1<br>2 ) contains innitely many<br>terms of the sequence.<br>I1 := fn 2 N : an 2 B(xio ;<br>1<br>2<br>)g:<br>I1 is innite and for any fangn2I1 , d(an; am) &lt; 1 8m; n 2 I1.<br>For 1<br>23 &gt; 0; 9×1; x2:::xr : A<br>[r<br>i=1<br>B(xi;<br>1<br>22 ) so, 9i1 2 f1; 2:::rg such that<br>B(xi1 ; 1<br>22 ) contains innitely many terms of the subsequence fangn2I1 .<br>Now dene I2 :=</p><p>n 2 I1 : an 2 B(xi1 ;<br>1<br>22 )</p><p>:<br>6<br>I2 I1 and for n;m 2 I2,<br>d(an; am) d(an; ai1) + d(am; ai1)<br>1<br>22 +<br>1<br>22 =<br>1<br>2<br>:<br>Iteratively, for any Ik innite we can get Ik+1 innite with Ik+1 Ik and<br>8m; n 2 Ik+1<br>d(an; am) &lt;<br>1<br>2k<br>Given nk choose nk+1 2 Ik+1 such that nk+1 &gt; nk (this is possible because Ik+1<br>is innite). Now for j &gt; k the index nj belongs toIk (becauseI1 I2 I3<br>:::is a nested sequence of sets)<br>Now for k &lt; j<br>d(ank ; anj ) d(ank ; ank+1) + : : : + d(anjô€€€1 ; anj )<br>1<br>2k + 1<br>2k+1 + : : : + 1<br>2jô€€€1<br>= 1<br>2k (1 + 1<br>2 + : : : + 1<br>2jô€€€kô€€€1 )<br>1<br>2kô€€€1 ô€€€! 0; as k ô€€€! 1<br>fankgk1 is a Cauchy subsequence of fang.Its convergence is guaranteed by the<br>completeness X. Hence A is precompact<br>Theorem 1.1.15. (Arzela-Ascoli)[6]<br>A subset A of the space of continuous functions C(K), where K is a non-empty<br>compact subset of RN, is relatively compact if and only if the two following con-<br>ditions are satised<br>i) A is uniformly bounded, i.e there exists M &gt; 0 such that 8x 2 X;<br>jf(x)j M; 8f 2 K.<br>ii) A is equicontinuous i.e 8 &gt; 0; 9 &gt; 0:<br>8 x; y 2 X; kx ô€€€ yk ) jf(x) ô€€€ f(y)j ; 8f 2 K<br>.<br>Denition 1.1.16. An inner product (or scalar product) on a vector space X<br>is a scalar valued function, h ; i on X X such that<br>i) for each y 2 Xfixed; the functional x 7! hx; yi is linear..<br>ii) hx; yi = hy; xi; where the bar the complex conjugation.<br>iii) hx; xi 0; 8 x 2 X.<br>iv) 8 x 2 X, hx; xi = 0 if and only if x = 0.<br>7<br>The pair (X; h ; i) is called Pre-hilbertian (or inner product) space.<br>The function kkX =<br>p<br>hx; xi denes a canonical norm on X.<br>Denition 1.1.17. A Pre-hilbertian space (H; hi) is called a Hilbert space if<br>it is complete when equipped with the corresponding canonical norm.<br>Denition 1.1.18. Cauchy-Schwarz Inequality and parallelogram<br>law<br>Let hx; yi be an inner product on a vector space X.<br>Then<br>jhx; yij kxkkyk for all x; y 2 X: (1.1.2)<br>kx + yk2 + kx ô€€€ yk2 = 2(kxk2 + kyk2) for all x; y 2 X (1.1.3)<br>Proposition 1.1.19. The polarization identity.<br>Let X be an inner product space. Then for arbitrary x; y 2 X,<br>hx; yi =<br>1<br>4<br>fjjx + yjj2 ô€€€ jjx ô€€€ yjj2 + ijjx + iyjj2 ô€€€ ijjx ô€€€ iyjj2g (1.1.4)<br>where i2 = ô€€€1.<br>Theorem 1.1.20. Jordan-Von Neumann.<br>The norm of a normed linear space X is given by an inner product if and only<br>if this norm satises the parallelogram law, i.e, if and only if,<br>kx + yk2 + kx ô€€€ yk2 = 2(kxk2 + kyk2); 8 x; y 2 X:<br>Denition 1.1.21. Let H be a Hilbert space and x; y 2 H. We say that x is<br>orthogonal to y denoted by x ? y if hx; yi = 0.<br>For M H, we say that x is orthogonal to M and write x ? M, if x is<br>orthogonal to every vector y in M.<br>The subset of vectors of H ortogonal to M is denoted by<br>M? = fx 2 H : x ? Mg<br>and is called the orthogonal complement of M in H.<br>Proposition 1.1.22. [1] Let M and N be arbitrary subspaces of a Hilbert<br>space H.Then the following holds<br>i) M? is a closed subspace of H<br>ii) M M??,<br>iii) if M N then N? M?<br>iv) (M?)? = M.<br>8<br>Denition 1.1.23. Given a closed subspace M of H, an operator P dened<br>on H is called the orthogonal projection onto M if<br>P(m + n) = m, for all m 2 M and n 2 M?<br>Theorem 1.1.24. The projection Theorem<br>Let H be a Hilbert space and M be a closed subspace of H. For an arbitrary<br>given vector x 2 H, there exists a unique vector m 2 M such that<br>jjx ô€€€ mjj jjx ô€€€ mjj for allm 2 M:<br>Furthermore, z 2 M is the unique vector m if and only if<br>(x ô€€€ z)?M:<br>Corollary 1.1.25. Direct Sum Decomposition<br>Let M be a closed subspace of a Hilbert Space, H. Then H = M M?:<br>Denition 1.1.26. An orthonormal system is a family f’igi2I of elements of<br>H such that h’i; ‘ji = i j , where, i j is the kronecker delta dened by<br>i j =</p><p>1 i = j;<br>0 i 6= j:<br>Example 1.1.27.</p><p>ei2nx ; n 2 Z</p><p>is an othonormal system for L2([0; 1])<br>It is easy to see that<br>hen; emi =<br>Z<br>[0;1]<br>ei2nxeô€€€i2mxdx<br>=<br>Z<br>[0;1]<br>ei2(nô€€€m)xdx =</p><p>1 if n = m<br>0 if n 6= m:<br>Denition 1.1.28. [1] A Hilbert space H is separable if H contains a countable<br>dense subset. Equivalently, a Hilbert space H is said to be separable if there<br>exists a sequence of vectors v1; v2; :::; vk; ::: which span a dense subspace of H.<br>Theorem 1.1.29.<br>A Hilbert space admits a countable orthonormal basis if and only if it is sepa-<br>rable.<br>Theorem 1.1.30. Riesz representation theorem<br>Let f be a continuous linear form on Hilbert space i.e f 2 H. Then there<br>exist a unique uf 2 H such that hf; vi = hv; uf i for all v 2 H.<br>Furthermore, we have kfkH = kufkH<br>9<br>Denition 1.1.31. Let X and Y be Banach spaces and T 2 B(X; Y ), dene<br>the dual (also called adjoint) operator as a map T : Y ô€€€! X dened by<br>Tf = f T<br>that is,<br>(Tf)(x) = f(T(x)) for all x 2 X:<br>T is called the (topological) dual or adjoint operator of T.<br>Remark. T 2 B(Y ;X)<br>Denition 1.1.32. Adjoint operators on Hilbert spaces<br>Let T 2 B(H1;H2), the adjoint of T is the unique map T : H2 ô€€€! H1 such<br>that<br>hTx; yi = hx; Tyi for all x 2 H1 and all y 2 H2<br>Example 1.1.33. Let be a bounded complex valued Lebesque measurable<br>function on [a; b]. Let<br>T : L2([a; b]) ô€€€! L2([a; b])<br>be the bounded linear operator dened by T(f) = f, that is,<br>(Tf)(t) = (t) f(t) for a:e: t 2 [a; b]:<br>For all f; g 2 L2([a; b]) we have<br>hTf; gi =<br>Z b<br>a<br>(Tf)(t)g(t)dt =<br>Z b<br>a<br>(t)f(t)g(t)dt = hf; gi:<br>Thus T(g) = g :<br>Theorem 1.1.34. Let T : L2([a; b]) ô€€€! L2([a; b]) be the bounded operator<br>dened by<br>(Tf)(t) =<br>Z b<br>a<br>G(t; s)f(s)ds<br>where G is in L2([a; b] [a; b]).<br>For all g 2 L2([a; b]),<br>(Tg)(t) =<br>Z b<br>a<br>G(s; t)g(s)ds :<br>Proof.<br>10<br>hTf; gi =<br>Z b<br>a<br>Z b<br>a<br>G(t; s)f(s)ds</p><p>g(t)dt<br>=<br>Z b<br>a<br>f(s)<br>Z b<br>a<br>G(t; s)g(t)dt</p><p>ds by Fubini’s thorem<br>= hf; gi<br>where<br>g(s) =<br>Z b<br>a<br>G(t; s)g(t)dt<br>1.2 Complexication of real Banach spaces<br>Many of the classical Banach functions spaces exist in real or complex-valued<br>versions. Examples are the Lp()-spaces and C(K)-spaces. Usually one is in-<br>terested in knowing whether a theory carried for real Banach spaces also holds<br>for complex Banach spaces (or vice-versa). An approach of solution is given by<br>the Complexication theory of real Banach spaces. Complexication preserves<br>norm and allows us to extend all basic notions on any arbitrary real Banach<br>space to Complex Banach space.<br>Denition 1.2.1. A complex vector space EC is a complexication of a real<br>vector space E if the two following conditions holds.<br>a) There is a one-to-one real linear map j : E ô€€€! EC<br>b) complex-span (j(E)) = EC.<br>There are, however various alternative concrete descriptions, some of which<br>include Ordered pair, Tensor and Linear operator descriptions of complexi-<br>cation.<br>Hence T is well dened ,infact T 2 B(`2)<br>Ordered pair description of a complexication.<br>If E is a real vector space, we can make E E a vector space by dening<br>(x; y) + (u; v) := (x + u; y + v) 8x; y; u; v 2 E<br>( + i)(x; y) := (x ô€€€ y; x + y) 8x; y 2 E; 8; 2 R:<br>11<br>Consider the map<br>j : E ô€€€! E E<br>x 7! (x; 0):<br>Clearly,<br>j(x + y) = j(x) + j(y) for any x; y 2 E and 2 R;<br>Ker(j) = fx 2 E : j(x) = (0; 0)g = f0g;<br>and<br>Complex ô€€€ span(j(E)) = E E:<br>The map j satises the conditions a) F(T(B)) ) F(T(B))andb)above; andsothiscomplexvectorspaceisacomplexificationofE:ItisconvenienttodenoteitbyEEiE:andalsosuppressrefrencetojbywrittingz = x+iy for the element z =<br>(x; y) = j(x) + ij(y):Itisnaturaltowritex=For other two descriptions, we refer to[4].<br>Denition 1.2.2. Let E be a real Banach space and EC := EiE.<br>ô€€€<br>EC; jj jj</p><p>is called a complexication of E if<br>ô€€€<br>EC; jjjj</p><p>is a complex Banach space, jjjjjE<br>is the original norm of E (i.e jjx + i0jj = jjxjj; 8x 2 E) and<br>jjx + iyjj = jjx ô€€€ iyjj 8x; y 2 E:<br>Now we might ask the Question: Is there a norm on EC which makes EC<br>a complex Banach space and induces the original norm onE ?<br>The answer is armative and there are innitely many ways to do so.[4].<br>Proposition 1.2.3. Let EC be a complexication of the real space E endowed<br>with a norm jj jj such that (E; jj jj) is Banach. Then jj jjT as dened below<br>denes a norm on EC.<br>jjx + iyjjT := sup<br>0t2<br>jj(cos t)x ô€€€ (sin t)yjj<br>All other complexication norms jj jj on EC are equivalent to jj jjT . Indeed<br>jjx + iyjjT jjx + iyjj 2jjx + iyjjT 8 x; y 2 E: (1.2.1)<br>Denition 1.2.4. Let E be a real Banach space. We say that a norm on the<br>complexication EC is reasonable if<br>c) jjj(x)jj = jjxjj 8x 2 E<br>d) jjx + iyjj = jjx ô€€€ iyjj x; y 2 E<br>When EC is equipped with such a norm, we call it a reasonable complexication<br>of E<br>12<br>Proposition 1.2.5. Let EC be a reasonable complexication of the real Banach<br>space E. for any x; y 2 E we have jjxjjE jjx+iyjjEC and jjyjjE jjx+iyjjEC<br>Proof. By property (c),<br>2jjxjjE = jj(x + iy) + (x ô€€€ iy)jjEC jjx + iyjjEC + jjx ô€€€ iyjjEC<br>An application of property d) gives jjxjjE jjx + iyjjEC<br>Similarly we have the other inequality.<br>Proposition 1.2.6. Let EC be a complexication of the real Banach space E<br>For any x; y 2 E we have,<br>sup<br>0t2<br>jj(cos t)x ô€€€ (sin t)yjjE jjx + iyjjEC<br>and<br>jjx + iyjjEC inf<br>0t2<br>(jj(cos t)x ô€€€ (sin t)yjjE + jj(sin t)x + (cos t)yjjE):<br>Proof. For each 0 t 2<br>jjx+iyjjEC = jjeit(x+iy)jjEC = jj((cos t)xô€€€(sin t)y)+i((sin t)x+(cos t)y)jjEC.<br>Using proposition (1.2.5) on the left and the triangle inequality on the right,<br>we have<br>jj(cos t)x ô€€€ (sin t)yjjE jjx + iyjjEC and jjx + iyjjEC jj(cos t)x ô€€€ (sin t)yjjE +<br>jj(sin t)x + (cos t)yjjE.<br>Hence, the result follows immediately.<br>Let us check verify that jj jjT is a reasonable complexication norm.<br>i)For any x 2 E, jjxjjT = jjx + i0jjT = sup<br>0t2<br>jjxcost ô€€€ 0sintjj = jjxjj<br>ii) jjxcost ô€€€ ysintjj = jjxcos(ô€€€t) + ysin(ô€€€t)jj and the function<br>t 7! xcost ô€€€ ysint is periodic with period of 2 for all x; y 2 E.Therefore,<br>jjx + iyjjT = sup<br>0t2<br>jjxcost ô€€€ ysintjj<br>= sup<br>t2R<br>jjxcos(ô€€€t) + ysin(ô€€€t)jj<br>= sup<br>t2R<br>jjxcos(t) + ysin(t)jj<br>= sup<br>0t2<br>jjxcost + ysintjj<br>jjx + iyjjT = jjx ô€€€ iyjjT<br>We have shown that property c) and d) are satised. Hence jjjjT is a reasonable<br>complexication norm.<br>Let us also verify the inequality in (1.2.1)<br>From proposition (1.2.6)<br>13<br>jjx + iyjjT jjx + iyjj inf<br>0t2<br>(jjxcost ô€€€ ysintjjE + jjxsint + ycostjjE)<br>sup<br>0t2<br>jjxcost ô€€€ ysintjjE + sup<br>0t2<br>jjxsint + ycostjjE<br>= 2jjx + iyjjT<br>jjx + iyjjT jjx + iyjj 2jjx + iyjjT<br>The norm jj jjT was rst considered by A.Y Taylor [ad]. (EC; jj jjT ) is known<br>as Taylor complexication of E.<br>There is a useful alternative description of jjx + iyjjT :<br>jjx + iyjjT = sup<br>0t2<br>jjxcost ô€€€ ysintjj<br>= sup<br>0t2<br>sup<br>jjfjjE1<br>jf(x)cost ô€€€ f(y)sintj<br>= sup<br>jjfjjE1<br>p<br>f(x)2 + f(y)2 ; 8x; y 2 E<br>Another feature of the Taylor complexication, is that it is a general complex-<br>ication whose denition is not tied to any specic characteristic of the real<br>Banach space E which is being complexied. Moreover, this procedure allows<br>us to extend continuous linear maps between real Banach space to complex<br>linear maps between their complexications without increasing the norm. If<br>L : E ô€€€! F is a linear map between real vector spaces E and F, there is a<br>unique complex-linear extension ~L : EC ô€€€! FC given by<br>~L<br>(x + iy) = L(x) + iL(y)<br>Proposition 1.2.7. Let E and F be real Banach spaces. If L 2 L(E; F), then<br>~L<br>2 L</p><p>(EC; jj jjT ); (FC; jj jjT )</p><p>and jj~ Ljj = jjLjj<br>Proof. Since ~L extends L, we have jj~Ljj jjLjj.<br>On the other hand, if x; y 2 E then<br>jj~L(x + iy)jjT = jjL(x) + iL(y)jjT = sup<br>0t2<br>jjL(x)cost ô€€€ L(y)sintjjF<br>= sup<br>0t2<br>jjL(xcost ô€€€ ysint)jjF<br>jjLjj sup<br>0t2<br>jjxcost ô€€€ ysintjjE<br>jj~L(x + iy)jjT jj~ Ljjjjx + iyjjT =) jj~Ljj jjLjj<br>Hence,<br>jj~Ljj = jjLjj<br>14<br>Taylor’s procedure is just one of innitely many procedures with similar<br>properties.<br>1.3 Some function spaces (Lp, Sobolev spaces)<br>Denition 1.3.1. Let 1 p &lt; 1;<br>be an open bounded subset of Rn. We<br>dene<br>Lp(<br>) as the set of measurable functions f :<br>ô€€€! R such that<br>Z</p><p>jf(x)jpdx &lt; +1<br>L1(<br>) as the set of measurable functions f :<br>ô€€€! R such that esupjfj &lt; 1<br>where,<br>esupjfj = inffk &gt; 0; jf(x)j k a:e x 2<br>g<br>For f 2 Lp(<br>), we dene,<br>jjfjjp =<br>Z</p><p>jf(x)jp<br>1<br>p ; 1 p &lt; 1:<br>jjfjj1 = esupjfj; if p = 1:<br>Theorem 1.3.2. The following properties holds for Lp space<br>i) Lp-space is Banach for 1 p 1<br>ii) Lp-space is Re exive for 1 &lt; p &lt; 1<br>iii) Lp-space is Separable for 1 p &lt; 1<br>F(T(B)) ) F(T(B))Itisalsoexpedienttorecallsomenotationsandbasicresultsfromdistributiontheory:<br>Denition 1.3.3. The Space L10<br>(<br>) is the space of all Lebesgue measurable<br>functions in<br>having absolute value integrable on each compact subset of</p><p>A multi-index is a vector (1; 2:::n) 2 Nn.<br>The length of is given by jj = 1 + ::: + n<br>We also dene the generalized derivative<br>D =<br>@jj<br>@1×1 : : :@nxn<br>Denition 1.3.4.<br>A locally integrable function v i.e element of L10<br>(<br>) is called the ô€€€ th weak<br>derivative of u 2 L10<br>(<br>), if it satises<br>Z</p><p>u(x)D(x)dx = (ô€€€1)jj<br>Z</p><p>v(x)(x)dx; 8 2 D(<br>)<br>Where D(<br>) denotes the set of C1-functions on<br>with compact support in</p><p>15<br>Let x 2 Rn, we write x = (x0; xn) with x0 2 Rnô€€€1, x0 = (x1; x2; :::xnô€€€1). We<br>consider the following notations[5].<br>Rn<br>+ = fx = (x0; xn) 2 Rn : xn &gt; 0g<br>B = fx = (x0; xn) 2 Rn : jjx0jj &lt; 1; jxnj &lt; 1g<br>B+ = fx = (x0; xn) 2 B : xn &gt; 0g<br>B0 = fx = (x0; xn) 2 B : xn = 0g<br>Denition 1.3.5. We say that an open subset<br>Rn is of class Cm(<br>)(m; integer)<br>if for every x 2 @<br>, there exist an open neighbourhood U of x in Rn and a<br>map : B ô€€€! U such that,<br>i) is a bijection<br>ii) 2 Cm( B; U);ô€€€1 2 Cm(U<br>; B)<br>iii) (B+) =<br>U; (B0) = @<br>U<br>Denition 1.3.6.<br>Let 1 p +1, m 2 N,. The Sobolev space Wm;p(<br>) is dened by<br>Wm;p(<br>) = fu 2 Lp(<br>) j Du 2 LP (<br>) for all jj mg<br>We shall be working with the case p = 2 .The Sobolev space Wm;p(<br>) are<br>denoted by Hm(<br>). H1<br>0(<br>) is the closure of D(<br>) in H1(<br>)<br>. Finally, we shall consider two important results which are very instrumental<br>to the application of spectral theorem of compact self adjoint operators to elliptic<br>partial dierential equations.<br>Proposition 1.3.7. Poincare Inequality<br>Let 1 p &lt; 1 and<br>a bounded open subset of RN. Then there exist a<br>constant C(<br>; p) such that<br>jjujjLp(<br>) CjjrujjLp(<br>) 8u 2 W1;p<br>0 (<br>):<br>If<br>is connected and satises a C1 boundary condition, then there exists a<br>constant C(<br>; p) such that<br>jju ô€€€ ujjLp(<br>) CjjrujjLp(<br>); 8u 2 W1;p(<br>)<br>u =<br>1<br>j<br>j<br>Z</p><p>u(x)dx<br>, is the mean value of u on</p><p>16<br>Denition 1.3.8.<br>Let E and F be two normed vector spaces such that E F.We say Ec ,! F is<br>compact embeddings if any bounded subset of E is precompact in F ,or equiv-<br>alently any bounded sequence of E has a subsequence that converges in F.<br>17<br>CHAPTER</p> <br><p></p>