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PROGRESS on the INVARIANT SUBSPACE PROBLEM Contents 1 PROGRESS ON THE INVARIANT SUBSPACE PROBLEM BEN GOLDMAN Contents 1. Introduction 2 1.1.FiniteDimensions 2 2. FundamentalsofFunctionalAnalysis 3 2.1. Banach Spaces 3 2.2. The Open Mapping Theorem 4 2.3. Compact Operators 5 3. SpectralTheoryandLinearMethods 7 3.1. Elementary Functional Calculus 7 3.2. Simple Invariant Subspace Case 8 3.3. Gelfand’s Spectral Radius Formula 9 3.4. Hilden’s Method 10 4. Lomonosov’sProofandNonlinearMethods 11 4.1. Schauder’s Theorem 11 4.2. Lomonosov’s Method 13 5. The Counterexample 14 5.1. Preliminaries 14 5.2.ConstructingtheNorm 16 5.3.TheRemainingLemmas 17 5.4. The Proof 21 6. Acknowledgements 24 References 24 Date: August, 2020. 1 2 BEN GOLDMAN 1. Introduction An important tool in working on any abstract space is to search for invariants, notions which do not change after a shift. The most elementary of these is the fixed point, an element which remains stationary after a transformation of space. The study of linear operators then introduces an invariant equipped with nonzero dimensionality, the eigenvector. In this paper, we endeavor to study how these simple principles can be extended to a more abstract notion, the invariant subspace. We begin with some very elementary, motivating examples. 1.1. Finite Dimensions. Let V,W be complex vector spaces. Definition 1.1.1. Recall that an operator, T V W ,islinearifforallv,µ V and ↵,β R we have that T ↵v βµ ↵T ∶ v →βT µ ∈ ∈ Definition 1.1.2. Recall that, v,isaneigenvectorofalinearoperator,T ,witheigen- ( + ) = ( ) + ( ) value, λ C,ifT v λv. Example 1. Consider the linear transformation, T 2 2,givenbyT x, y ∈ ( ) = R R 2x, 3y .Thishaseigenvectors,1, 0 and 0, 1 with real eigenvalues, 2 and 3, respectively. ∶ → ( ) = ( ) ( ) ( ) Example 2. Now, consider the linear transformation, T R2 R2,whichrotatesthe plane by some angle, ✓.ItisfairlyeasytoobservethatT has no real eigenvalues. However, one observes that the unit disk, D v R2 ∶ v → 1 ,isinvariantinT (i.e. T D D). This motivates our definition, below: = { ∈ = } Definition 1.1.3. A subspace, W V ,isinvariantinalinearoperator,T V V ,if ( ) = T W W . ⊆ ∶ → Remark. Granted, not all invariant subspaces are created equal. For instance, the ( ) ⊆ origin in V is invariant under any linear transformation, but this does not tell us anything significant about the operator nor the space. Similarly, for any surjective linear operator, T V V ,wehavethatT V V ,whichisanotherunimportant example. We dub these the trivial invariant subspaces for an operator. ∶ → ( ) = Definition 1.1.4. The orbit of a vector, v V ,withrespecttoalinearoperator, T V V ,isgivenby ∈ O v,Tv,T 2v,T 3v,T 4v,...,T nv,... ∶ → v One notes that O is also an invariant subspace, but a fairly disappointing one. v = { } Remark. In order to avoid the aforementioned simple cases, we state the Invariant Subspace Problem: Under which conditions does T have a nontrivial, closed invariant subspace? PROGRESS ON THE INVARIANT SUBSPACE PROBLEM 3 It is fairly easy to prove this for the case of a finite dimensional complex vector space. Theorem 1.1.5. Any nonzero operator on a finite dimensional, complex vector space, V , admits an eigenvector. Proof. [A16] Let n dim V and suppose T V V is a nonzero linear oper- n ator. Select nonzero v V so that by finite dimensionality, we know that Ov v,Tv,T 2v,...,T nv =must( be) a linearly dependent∶ set.→ Thus, ∈ = ↵ T nv ↵ T n 1v ...↵ Tv ↵ v ⌘ T λ T λ ... T λ v 0 { n n 1 } 1 0 1 2 n − where we use λ − to denote λ I (the identity map). At least one of these factors + i C + i + = ( − )( − ) ( − ) = must have a nontrivial kernel which suffices to prove the theorem. One can adapt the proof above to∈ real valued vector spaces, but the irreducible polynomials end up being quadratic instead. ⇤ Corollary 1.1.6. Any nonzero operator on a finite dimensional, complex vector space, V , admits a closed, nontrivial invariant subspace. 2. Fundamentals of Functional Analysis 2.1. Banach Spaces. Definition 2.1.1. Recall that a Banach Space is a complete Vector Space with norm, . Lemma 2.1.2 (Riesz). Let x E, x 1 be the unit sphere in an infinite dimensional Banach Space, E, . .IfY E is a closed proper subspace of E then there exists x x " suchB = that{ ∈ = } ∗ ∗ ( ) 1 ⊂ = ( ) ∈ B y x y Y," 0 ∗ 1 " Proof. Pick an arbitrary x − so> that d ∀ inf∈ > x y . Select y so that + y Y ∗ x y d d"0 d d" and set ∈ ∈ B = − ∗ 1 1 − = + <x + ,x x y x y ∗ ∗ x y ∗ d d"0 ∗ We then have that ∈ B = ∗ ( − ) = ( − ) − + 1 1 1 x y x y d d"0 y inf x yˆ yˆ Y ∗ d d"0 ∗ d d"0 1 " wherey ˆ y − d =d" y − − ( + ) > ∈ ( − ) > 0+ + + ⇤ ∗ Theorem= 2.1.3+ ( (Riesz)+ ) . If E, . is an infinite dimensional banach space then the unit sphere, , is not compact. ( ) B 4 BEN GOLDMAN Proof. Pick x1 and set Y1 span x1 . Apply the previous lemma to yield x2 1 such that infy Y1 x2 y 2 and let Y2 span x1,x2 .Continuethisprocess ∈ B = { } inductively to∈ yield a sequence, xn and Yn ,suchthatxn has no convergent subsequence. − > = { } ⇤ { } { } This stands in stark contrast to finite dimensional vector spaces which have compact unit spheres. We conclude this subsection with a fact about linear operators on Banach Spaces. Definition 2.1.4. We define a norm on the space of linear operators between Banach spaces, T X, . X Y, . Y ,by ∶ ( ) → ( ) T sup T x Y x One notes that there is not necessarily = a∈B vector, ( )x0 ,forwhich T x0 T due to the loss of compactness. ∈ B ( ) = Theorem 2.1.5. If T X, . X Y, . Y is a linear map between Banach Spaces then it is continuous if and only if T ∶ ( ) → ( ) " Proof. If T then for all " 0if <x+∞y X T then < +∞ > −1 < T x T y Y x y X T x y T x y X " x y X Y ( ) − ( ) = − ( − ) ≤ − < We will use the manipulation above i.e. −T x Y T x X quite often. If T is 1 unbounded then for each n N select xn X such that xn X n and T xn Y n. ( ) ≤ This produces a sequence, xn 0suchthat T xn Y 1foralln. And thus, lim supx 0 T x Y 1, and∈ so T is not continuous∈ at the origin. = ( ) > ⇤ → ( ) ≥ → ( ) ≥ 2.2. The Open Mapping Theorem. Theorem 2.2.1 (Baire Category Theorem). Complete metric spaces are Baire spaces. More precisely, if X,⇢ is a complete metric space and n n N is a countable col- lection of open dense subsets then the intersection, , is also dense. [R10] n N n∈ ( ) {O } ∈ We require the above theorem from non-functionalI analysis= O in order to justify the following results. Theorem 2.2.2 (Open Mapping Theorem). If T X, . X Y, . Y is a surjec- tive, continuous linear map then T is an open map. ∶ ( ) → ( ) PROGRESS ON THE INVARIANT SUBSPACE PROBLEM 5 Proof. [R10] Let BX BX 0, 1 be the unit ball in X and BY be the unit ball in Y .ObservethatX r N rBX and that Y r N T rBX .BytheBaireCategory = ( ) Theorem, there exists R∈ N such that T RBX ∈ T BX 0,R has nonempty inte- rior. = ( ) = ( ) ∈ ( ) = ( ( )) Pick BY y0,r0 T BX 0,R so that for all y BY we have that y0 r0y BY y0,r0 . Thus, ( ) ⊆ ( r0(y ))T RBX T RBX∈ T 2RBX + ∈ ( ) and we thus have that B T R B where R 2R r . Y∈ ( X) − ( ) ⊆ ( 0 ) ∗ ∗ For any y Y and " 0,⊆ we( find a) point x =R BX such that T x y ". R We can refine this approach to get a stronger result.∗ Select x1 B 0, for which 2∗ ∈ 1 > R ∈ ( ) ( 1) − < Tx1 y and xn B 0, n such that T x1 x2 xn y n .Letting 2 2 ∗ ∈ 2 x n N xn where x n N xn 2R gives that T x y.Thus,wehave 1 shown ∗ − that < B T 2R∈∗B and so B 0, ∗ ( T+ B +.⋅⋅⋅+∗ ) − < ∈ Y X ∈ Y 2R X = ∑ ∗ ≤ ∑ ≤ ∗ ( ) = Finally, one shows⊆ ( that T )is open. Since T is ⊆ linear,( it) suffices to prove that the image of the unit ball is open (other open sets are achieved by translations/scalings). Take y T BX and select x BX such that Tx y.Pick" 0suchthat BX x, " BX and pick r sufficiently small so that BY y,r T BX x, " so that B y,r ∈T B( X .) ∈ = > ⇤ ( ) ⊆ ( ) ⊆ ( ( )) We finish this subsection with a handful of useful statements that are immediate ( ) ⊆ ( ) consequences of the Open Mapping Theorem. Corollary 2.2.3. If T X, . X Y, . Y is a bijective, continuous operator between two banach spaces then T 1 is continuous. [R10] ∶ ( −) → ( ) Corollary 2.2.4. If . 1 and . 2 are two norms on a Banach space, X, such that . 1 C . 2 then they are equivalent. [R10] ≤ 2.3. Compact Operators. Notation 2.3.1. Let X be a norm space. We call X the space of continuous linear operators mapping a space, X, into itself. If T X then T X X. L( ) For obvious reasons, we will be concentrating our e↵orts on these types of operators ∈ L( ) ∶ → for the remainder of the paper. The loss of compactness in Banach spaces makes it difficult to study these linear operators to the same degree as one could in a Euclidean setting. However, we define a special class of operators which has very nice properties (and that we will see, later on, satisfies the invariant subspace problem).
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