We define a truncated composition operator on the spaces P_n of n-degree polynomials with complex coefficients. After doing so, we concern ourselves with the complex symmetry of such operators, that is, whether there is an orthonormal basis that gives them a symmetric matrix representation

Jansen, Ruth

Rousseau, Rebecca K

English

Rousseau, Rebecca K.

Pillars at Taylor University

Taylor University Pillars at Taylor University Mathematics Student Scholarship Mathematics Department 2017 Complex Symmetry of Truncated Composition Operators Ruth Jansen Rebecca K. Rousseau Follow this and additional works at: https://pillars.taylor.edu/mathstudentscholarship Rose-Hulman Undergraduate Mathematics Journal Volume 18 Issue 1 Article 3 Complex Symmetry of Truncated Composition Operators Ruth Jansen Taylor University Rebecca K. Rousseau Taylor University Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Jansen, Ruth and Rousseau, Rebecca K. (2017) "Complex Symmetry of Truncated Composition Operators," Rose-Hulman Undergraduate Mathematics Journal: Vol. 18 : Iss. 1 , Article 3. Available at: https://scholar.rose-hulman.edu/rhumj/vol18/iss1/3 Rose-HulmanUndergraduateMathematicsJournalSponsored byRose-Hulman Institute of TechnologyDepartment of MathematicsTerre Haute, IN 47803mathjournal@rose-hulman.eduscholar.rose-hulman.edu/rhumjcomplex symmetry of truncatedcomposition operatorsRuth Jansena Rebecca K. Rousseau bVolume 18, No. 1, Spring 2017aTaylor UniversitybTaylor UniversityRose-Hulman Undergraduate Mathematics JournalVolume 18, No. 1, Spring 2017complex symmetry of truncatedcomposition operatorsAbstract.We define a truncated composition operator on the spaces Pn of n-degreepolynomials with complex coefficients. After doing so, we concern ourselves withthe complex symmetry of such operators, that is, whether there is an orthonormalbasis that gives them a symmetric matrix representation.Acknowledgements: We are greatly indebted to the work and advice of Carl Cowen,Stephan Garcia, and James Tener. We also wish to express our gratitude to Taylor Univer-sity’s Faculty Mentored Undergraduate Scholarship (FMUS) program, which funded thisresearch. Lastly, we would like to thank our faculty mentor Dr. Derek Thompson for pro-viding guidance in our research.Page 44 RHIT Undergrad. Math. J., Vol. 18, No. 11 IntroductionSymmetric matrices, whose entries form a mirror image across the diagonal, are a familiarconcept in the study of linear algebra and its applications. The concept of a complex sym-metric operator is similar in nature and has been shown to have a wide variety of interestingapplications within operator theory (Garcia, Putinar [2]). An operator is complex symmetricif it has a symmetric matrix representation with respect to some orthonormal basis. Thisis distinct from a self-adjoint matrix, in which entries across the diagonal are the complexconjugate of one another rather than identical.In recent years, researchers have shown an interest in the connection between complexsymmetry and composition operators (Narayan et. al.[4]). On any space of analytic functions,the composition operator with symbol Cϕ can be defined by Cϕf = f ◦ ϕ. Compositionoperators are most commonly studied on H2, the Hilbert space of analytic functions f(z) =∑∞k=0 akzk from the open unit disk D to C where ||f ||2 =∑∞k=0 |ak|2 <∞ (square-summableTaylor series). To make such operators work in finite dimensions, we consider them on thepolynomial subspace Pn of H2: the space of all polynomials of degree n or less with complexcoefficients. To make a composition operator with any given symbol map Pn back into Pn,we view the operator as map from Pn to H2, and then project the image back onto thepolynomial space Pn by removing all terms of degree n + 1 or higher (truncation).We write PnCϕ for the operator that maps Pn to itself and call it a truncated compositionoperator. This is equivalent to considering the upper-left block of the matrix representationof Cϕ on the Hardy space H2 in its standard basis. In particular, the traditional dot productused in Rn aligns with the inner product that Pn inherits as a subspace of H2, allowing us touse familiar tools, and to study complex symmetry in a finite-dimensional setting, yet withimplications for study on infinite-dimensional function spaces. Although our work couldtechnically be defined for a variety of symbols ϕ so that Cϕ acts on Pn, our ultimate goal isto inform further work on the Hardy space H2. To that end, we will focus on symbols thatare self-maps of D.For example, let ϕ(z) = z2−z =z2+ z24+ z38+ . . . and f(z) = z2. Then Cϕf will result inthe following Taylor series.(Cϕf)(z) = (f ◦ ϕ)(z) = (ϕ2)(z) =14z2 +28z3 +316z4 +432z5 +564z6 + · · ·This means that Cϕ acts as a map from Pn to H2; the operator Pn will truncate thehigher-degree terms and define PnCϕ as a self-map of Pn. Here, we have(PnCϕf)(z) = (Pnϕ2)(z) =14z2 +28z3 + · · ·+ n− 12nzn.Let ϕ : D→ D be some analytic function with Taylor series∑∞k=0 akzk. With respect tothe canonical basis {1, z, z2, z3, · · · , zn}, PnCϕ can be written as an (n+ 1)× (n+ 1) matrix.The columns of this matrix correspond to PnCϕ acting on each element of the basis. Figure1 shows P2Cϕ when ϕ(z) =z2−z .RHIT Undergrad. Math. J., Vol. 18, No. 1 Page 451 0 00 1200 1414Figure 1: Matrix representation of P2CϕThis matrix would also represent the upper left hand corner of the infinite matrix repre-sentation of Cϕ with respect to the same canonical basis (now with infinitely many elements).Next, we define a complex symmetric operator and give an example of a truncated com-position operator exhibiting this property.Definition. An operator T is said to be a complex symmetric operator (CSO) if there isan orthonormal basis with respect to which T has a symmetric matrix representation (whichmay contain complex entries).As an example of a complex symmetric operator, we again consider ϕ(z) = z2−z . Withrespect to the orthonormal basis{1,(1 +√2)e3iπ8√4 + 2√2z +e3iπ8√4 + 2√2z2,(1−√2)e−iπ8√4− 2√2z +e−iπ8√4− 2√2z2},P2Cϕ has the following symmetric matrix representation:1 0 00 18(3 +√2) i80 i818(3−√2)Note that this matrix is symmetric, but not self-adjoint. That is, the complex entries areidentical over the diagonal and not the complex conjugate of one another.In the remainder of this paper, we will explore the question of when a truncated com-position operator is complex symmetric. In Section 2, we describe our methodology andintroduce the Strong Angle Test, which will be used throughout the remainder of the paper.We will then explore the relationship between the complex symmetry of Cϕ and P2Cϕ forvarious symbols ϕ(z). Finally, we will determine symbols ϕ(z) for which PnCϕ is representedby a lower triangular matrix in Section 3, and for which P2Cϕ is represented by an uppertriangular matrix in Section 4, with respect to the canonical basis. To conclude, we willpresent in Section 5 a few questions for further consideration in the future.2 Methodology and Initial FindingsIt is often difficult to find the correct orthonormal basis in order to determine if an operatoris complex symmetric. The following theorem allows us to determine whether such a basisexists using just one matrix representation of our operator.Page 46 RHIT Undergrad. Math. J., Vol. 18, No. 1Theorem (The Strong Angle Test [1]). Given an n × n matrix M , let u1,u2...un be theunit eigenvectors of M and let v1,v2...vn be the corresponding unit eigenvectors of M∗, theconjugate transpose of M . Then M is a CSO if and only if〈ui,uj〉〈uj,uk〉〈uk,ui〉 = 〈vi,vj〉〈vj,vk〉〈vk,vi〉for all 1 ≤ i ≤ j ≤ k ≤ n.Example. Let ϕ(z) = z2−z . With respect to the basis {1, z, z2}, P2Cϕ has the followingmatrix representation and eigenvectors: 1 0 00 1200 1414u1 =100u2 = 01√21√2u3 =001v1 =100v2 =010v3 = 0− 1√21√2 .From these eigenvectors, we have the following:〈u1,u2〉〈u2,u3〉〈u3,u1〉 = 0×1√2× 0 = 0〈v1,v2〉〈v2,v3〉〈v3,v1〉 = 0×−1√2× 0 = 0.Thus, P2Cϕ is a complex symmetric operator by the Strong Angle Test.We wish to examine whether the property of complex symmetry carries over from Cϕ toPnCϕ. We summarize the results from examining the case of P2Cϕ for 4 different definitionsof ϕ. The table below displays the results of testing P2Cϕ using the Strong Angle Testalongside knowledge of Cϕ from the work in Narayan et.al. [4]. The symbol Xindicates thatthe operator is a complex symmetric operator for the given ϕ and × indicates that it is not.Note that the complex symmetry of Cϕ seems to have no bearing on the complex symmetryof P2Cϕ, one way or the other.Although we found some truncated composition operators that were complex symmetric,our main result was that for the symbols we considered, complex symmetry does not holdfor any particular symbol ϕ as the dimension of the truncation was increased. In the nextsections, we give our justification for this result, considering only symbols that induce upper-and lower-triangular matrix representations. This was inspired by our knowledge of the workof Narayan et. al. [4], which only considered symbols with this property. In every case, weconsider symbols whose Taylor series have only real coefficients.RHIT Undergrad. Math. J., Vol. 18, No. 1 Page 47ϕ P2Cϕ Cϕ12z + 12× ×13z + 13× Xz2−z X ×z3−z X XTable 1: Complex Symmetry of Operators given ϕ3 The Lower Triangular Case3.1 Examining P2CϕFor ϕ(z) =∑∞k=0 akzk with a0 = 0, the matrix representation of P2Cϕ is lower triangular.1 0 00 a1 00 a2 a12Figure 2: Matrix of P2Cϕ with a0 = 0Here, f(z) ≡ 1 is a unit eigenvector for both P2Cϕ and (P2Cϕ)∗, while all of the otherunit eigenvectors for both operators do not have a constant term. Therefore, for all a1 anda2, 〈u1,u2〉 = 〈v1,v2〉 = 0. So, P2Cϕ is always a complex symmetric operator by the StrongAngle Test. When will symbols of the same form produce complex symmetric PnCϕ forlarger values of n?3.2 Examining PnCϕ, n > 2For further study, we consider PnCϕ where ϕ(z) = a1z + a2z2. For the following, we takeadvantage of the fact that for PnCϕ, 〈u1,ui〉 = 〈v1,vi〉 = 0 for all 2 ≤ i ≤ n + 1.Page 48 RHIT Undergrad. Math. J., Vol. 18, No. 1Figure 3: Values of a1 and a2 for which P3Cϕ is a CSO.The graph in Figure 3 shows values of a1 and a2 for which P3Cϕ with ϕ(z) = a1z + a2z2is complex symmetric by the Strong Angle Test. The shaded region represents symbols thatmap the open unit interval to itself (those in which |a1| + |a2| < 1), which is our area ofinterest.Figure 4: The Strong Angle Test on P4CϕThe four equations produced by applying the Strong Angle Test to P4Cϕ are graphed inFigure 4. There are at most two points satisfying all 4 equations when both coefficients arenon-zero.RHIT Undergrad. Math. J., Vol. 18, No. 1 Page 49Figure 5: A closer view of Figure 4 along with Figure 3In Figure 5, it appears that the four curves intersect near a1 = .569 and a2 = .356. Theblack line represents the curve plotted in Figure 3, which does not pass near this point. So,if there is a function ϕ(z) = a1z + a2z2 for which P4Cϕ is complex symmetric, the previoustruncation P3Cϕ was not.Figure 6: The Strong Angle Test on P5CϕBy applying the same approach to P5Cϕ, the resulting system of equations has no solution.This can be seen in Figure 6. Therefore, there is no ϕ(z) = a1z + a2z2 for which P5Cϕ iscomplex symmetric.In summary, it appears that there is no ϕ(z) = a1z + a1z2 for which PnCϕ is a complexsymmetric operator for all n. Moreover, we seem to lose a dimension in our solution set foreach larger value of n. It seems that by forcing each truncation to be in terms of the samecoefficients, we are limiting ourselves too far. In order to preserve complex symmetry fromPage 50 RHIT Undergrad. Math. J., Vol. 18, No. 1one truncation to the next, perhaps ϕ needs to introduce new coefficients into the corre-sponding matrix.Conjecture. If ϕ has a finite Taylor series, then there exists n such that PnCϕ is not acomplex symmetric operator.4 Upper Triangular Case, P2CϕFor ϕ(z) =∑∞k=0 akzk with a2 = 0, the matrix representation of P2Cϕ is upper triangular.1 a0 a020 a1 2a0a10 0 a12Figure 7: Matrix of P2Cϕ with a2 = 0If a0 = 0, the matrix above will be diagonal, and therefore symmetric. Other than thistrivial solution, there are no real values of a0 and a1 for which P2Cϕ is complex symmetricby the Strong Angle Test.5 Further QuestionsWe conclude with some questions for further research.1. For ϕ(z)=a1z, PnCϕ is a CSO for all n because the corresponding matrix will alwaysbe diagonal. Besides this trivial case, are there any symbols ϕ for which PnCϕ is aCSO for all n?2. Composition operators on H2 that are complex symmetric have yet to be completelyclassified, particularly in the difficult case when ϕ is not linear-fractional. Can trun-cated composition operators help solve this problem?3. What other properties beyond complex symmetry, meaningful in both finite and infinitedimensions, can be studied for truncated composition operators?References[1] Levon Balayan and Stephan Ramon Garcia. (2010). Unitary Equivalence to a ComplexSymmetric Matrix: Geometric Criteria. Operators and Matrices, 4(1): 53-76.[2] Stephen Ramon Garcia and Mihai Putinar. (2005). Complex Symmetric Operators andApplications. Transactions of the American Mathematical Society 358(3): 1285-1315.RHIT Undergrad. Math. J., Vol. 18, No. 1 Page 51[3] Carl Cowen and Barbara MacCluer. (1995). Composition Operators on Spaces of AnalyticFunctions. CRC Press, Inc.[4] Sivaram K. Narayan, Daniel Sievewright, and Derek Thompson (2016). Complex Symmet-ric Composition Operators on H2. Journal of Mathematical Analysis and Applications,443(1): 625-630.[5] James E. Tener. (2010). Unitary Equivalence to a Complex Symmertric Matrix. PomonaCollege, Claremont, CA.

Complex Symmetry of Truncated Composition Operators

Rose-Hulman Institute of Technology: Rose-Hulman Scholar

Rose-Hulman Undergraduate Mathematics Journal Volume 18 Issue 1 Article 3 Complex Symmetry of Truncated Composition Operators Ruth Jansen Taylor University Rebecca K. Rousseau Taylor University Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Jansen, Ruth and Rousseau, Rebecca K. (2017) "Complex Symmetry of Truncated Composition Operators," Rose-Hulman Undergraduate Mathematics Journal: Vol. 18 : Iss. 1 , Article 3. Available at: https://scholar.rose-hulman.edu/rhumj/vol18/iss1/3 Rose-HulmanUndergraduateMathematicsJournalSponsored byRose-Hulman Institute of TechnologyDepartment of MathematicsTerre Haute, IN 47803mathjournal@rose-hulman.eduscholar.rose-hulman.edu/rhumjcomplex symmetry of truncatedcomposition operatorsRuth Jansena Rebecca K. Rousseau bVolume 18, No. 1, Spring 2017aTaylor UniversitybTaylor UniversityRose-Hulman Undergraduate Mathematics JournalVolume 18, No. 1, Spring 2017complex symmetry of truncatedcomposition operatorsAbstract.We define a truncated composition operator on the spaces Pn of n-degreepolynomials with complex coefficients. After doing so, we concern ourselves withthe complex symmetry of such operators, that is, whether there is an orthonormalbasis that gives them a symmetric matrix representation.Acknowledgements: We are greatly indebted to the work and advice of Carl Cowen,Stephan Garcia, and James Tener. We also wish to express our gratitude to Taylor Univer-sity’s Faculty Mentored Undergraduate Scholarship (FMUS) program, which funded thisresearch. Lastly, we would like to thank our faculty mentor Dr. Derek Thompson for pro-viding guidance in our research.Page 44 RHIT Undergrad. Math. J., Vol. 18, No. 11 IntroductionSymmetric matrices, whose entries form a mirror image across the diagonal, are a familiarconcept in the study of linear algebra and its applications. The concept of a complex sym-metric operator is similar in nature and has been shown to have a wide variety of interestingapplications within operator theory (Garcia, Putinar [2]). An operator is complex symmetricif it has a symmetric matrix representation with respect to some orthonormal basis. Thisis distinct from a self-adjoint matrix, in which entries across the diagonal are the complexconjugate of one another rather than identical.In recent years, researchers have shown an interest in the connection between complexsymmetry and composition operators (Narayan et. al.[4]). On any space of analytic functions,the composition operator with symbol Cϕ can be defined by Cϕf = f ◦ ϕ. Compositionoperators are most commonly studied on H2, the Hilbert space of analytic functions f(z) =∑∞k=0 akzk from the open unit disk D to C where ||f ||2 =∑∞k=0 |ak|2 <∞ (square-summableTaylor series). To make such operators work in finite dimensions, we consider them on thepolynomial subspace Pn of H2: the space of all polynomials of degree n or less with complexcoefficients. To make a composition operator with any given symbol map Pn back into Pn,we view the operator as map from Pn to H2, and then project the image back onto thepolynomial space Pn by removing all terms of degree n + 1 or higher (truncation).We write PnCϕ for the operator that maps Pn to itself and call it a truncated compositionoperator. This is equivalent to considering the upper-left block of the matrix representationof Cϕ on the Hardy space H2 in its standard basis. In particular, the traditional dot productused in Rn aligns with the inner product that Pn inherits as a subspace of H2, allowing us touse familiar tools, and to study complex symmetry in a finite-dimensional setting, yet withimplications for study on infinite-dimensional function spaces. Although our work couldtechnically be defined for a variety of symbols ϕ so that Cϕ acts on Pn, our ultimate goal isto inform further work on the Hardy space H2. To that end, we will focus on symbols thatare self-maps of D.For example, let ϕ(z) = z2−z =z2+ z24+ z38+ . . . and f(z) = z2. Then Cϕf will result inthe following Taylor series.(Cϕf)(z) = (f ◦ ϕ)(z) = (ϕ2)(z) = 14z2 +28z3 +316z4 +432z5 +564z6 + · · ·This means that Cϕ acts as a map from Pn to H2; the operator Pn will truncate thehigher-degree terms and define PnCϕ as a self-map of Pn. Here, we have(PnCϕf)(z) = (Pnϕ2)(z) =14z2 +28z3 + · · ·+ n− 12nzn.Let ϕ : D→ D be some analytic function with Taylor series ∑∞k=0 akzk. With respect tothe canonical basis {1, z, z2, z3, · · · , zn}, PnCϕ can be written as an (n+ 1)× (n+ 1) matrix.The columns of this matrix correspond to PnCϕ acting on each element of the basis. Figure1 shows P2Cϕ when ϕ(z) =z2−z .RHIT Undergrad. Math. J., Vol. 18, No. 1 Page 451 0 00 1200 1414Figure 1: Matrix representation of P2CϕThis matrix would also represent the upper left hand corner of the infinite matrix repre-sentation of Cϕ with respect to the same canonical basis (now with infinitely many elements).Next, we define a complex symmetric operator and give an example of a truncated com-position operator exhibiting this property.Definition. An operator T is said to be a complex symmetric operator (CSO) if there isan orthonormal basis with respect to which T has a symmetric matrix representation (whichmay contain complex entries).As an example of a complex symmetric operator, we again consider ϕ(z) = z2−z . Withrespect to the orthonormal basis{1,(1 +√2)e3ipi8√4 + 2√2z +e3ipi8√4 + 2√2z2,(1−√2)e−ipi8√4− 2√2z +e−ipi8√4− 2√2z2},P2Cϕ has the following symmetric matrix representation:1 0 00 18(3 +√2) i80 i818(3−√2)Note that this matrix is symmetric, but not self-adjoint. That is, the complex entries areidentical over the diagonal and not the complex conjugate of one another.In the remainder of this paper, we will explore the question of when a truncated com-position operator is complex symmetric. In Section 2, we describe our methodology andintroduce the Strong Angle Test, which will be used throughout the remainder of the paper.We will then explore the relationship between the complex symmetry of Cϕ and P2Cϕ forvarious symbols ϕ(z). Finally, we will determine symbols ϕ(z) for which PnCϕ is representedby a lower triangular matrix in Section 3, and for which P2Cϕ is represented by an uppertriangular matrix in Section 4, with respect to the canonical basis. To conclude, we willpresent in Section 5 a few questions for further consideration in the future.2 Methodology and Initial FindingsIt is often difficult to find the correct orthonormal basis in order to determine if an operatoris complex symmetric. The following theorem allows us to determine whether such a basisexists using just one matrix representation of our operator.Page 46 RHIT Undergrad. Math. J., Vol. 18, No. 1Theorem (The Strong Angle Test [1]). Given an n × n matrix M , let u1,u2...un be theunit eigenvectors of M and let v1,v2...vn be the corresponding unit eigenvectors of M∗, theconjugate transpose of M . Then M is a CSO if and only if〈ui,uj〉〈uj,uk〉〈uk,ui〉 = 〈vi,vj〉〈vj,vk〉〈vk,vi〉for all 1 ≤ i ≤ j ≤ k ≤ n.Example. Let ϕ(z) = z2−z . With respect to the basis {1, z, z2}, P2Cϕ has the followingmatrix representation and eigenvectors: 1 0 00 1200 1414u1 =100u2 = 01√21√2u3 =001v1 =100v2 =010v3 = 0− 1√21√2 .From these eigenvectors, we have the following:〈u1,u2〉〈u2,u3〉〈u3,u1〉 = 0× 1√2× 0 = 0〈v1,v2〉〈v2,v3〉〈v3,v1〉 = 0×− 1√2× 0 = 0.Thus, P2Cϕ is a complex symmetric operator by the Strong Angle Test.We wish to examine whether the property of complex symmetry carries over from Cϕ toPnCϕ. We summarize the results from examining the case of P2Cϕ for 4 different definitionsof ϕ. The table below displays the results of testing P2Cϕ using the Strong Angle Testalongside knowledge of Cϕ from the work in Narayan et.al. [4]. The symbol Xindicates thatthe operator is a complex symmetric operator for the given ϕ and × indicates that it is not.Note that the complex symmetry of Cϕ seems to have no bearing on the complex symmetryof P2Cϕ, one way or the other.Although we found some truncated composition operators that were complex symmetric,our main result was that for the symbols we considered, complex symmetry does not holdfor any particular symbol ϕ as the dimension of the truncation was increased. In the nextsections, we give our justification for this result, considering only symbols that induce upper-and lower-triangular matrix representations. This was inspired by our knowledge of the workof Narayan et. al. [4], which only considered symbols with this property. In every case, weconsider symbols whose Taylor series have only real coefficients.RHIT Undergrad. Math. J., Vol. 18, No. 1 Page 47ϕ P2Cϕ Cϕ12z + 12× ×13z + 13× Xz2−z X ×z3−z X XTable 1: Complex Symmetry of Operators given ϕ3 The Lower Triangular Case3.1 Examining P2CϕFor ϕ(z) =∑∞k=0 akzk with a0 = 0, the matrix representation of P2Cϕ is lower triangular.1 0 00 a1 00 a2 a12Figure 2: Matrix of P2Cϕ with a0 = 0Here, f(z) ≡ 1 is a unit eigenvector for both P2Cϕ and (P2Cϕ)∗, while all of the otherunit eigenvectors for both operators do not have a constant term. Therefore, for all a1 anda2, 〈u1,u2〉 = 〈v1,v2〉 = 0. So, P2Cϕ is always a complex symmetric operator by the StrongAngle Test. When will symbols of the same form produce complex symmetric PnCϕ forlarger values of n?3.2 Examining PnCϕ, n > 2For further study, we consider PnCϕ where ϕ(z) = a1z + a2z2. For the following, we takeadvantage of the fact that for PnCϕ, 〈u1,ui〉 = 〈v1,vi〉 = 0 for all 2 ≤ i ≤ n + 1.Page 48 RHIT Undergrad. Math. J., Vol. 18, No. 1Figure 3: Values of a1 and a2 for which P3Cϕ is a CSO.The graph in Figure 3 shows values of a1 and a2 for which P3Cϕ with ϕ(z) = a1z + a2z2is complex symmetric by the Strong Angle Test. The shaded region represents symbols thatmap the open unit interval to itself (those in which |a1| + |a2| < 1), which is our area ofinterest.Figure 4: The Strong Angle Test on P4CϕThe four equations produced by applying the Strong Angle Test to P4Cϕ are graphed inFigure 4. There are at most two points satisfying all 4 equations when both coefficients arenon-zero.RHIT Undergrad. Math. J., Vol. 18, No. 1 Page 49Figure 5: A closer view of Figure 4 along with Figure 3In Figure 5, it appears that the four curves intersect near a1 = .569 and a2 = .356. Theblack line represents the curve plotted in Figure 3, which does not pass near this point. So,if there is a function ϕ(z) = a1z + a2z2 for which P4Cϕ is complex symmetric, the previoustruncation P3Cϕ was not.Figure 6: The Strong Angle Test on P5CϕBy applying the same approach to P5Cϕ, the resulting system of equations has no solution.This can be seen in Figure 6. Therefore, there is no ϕ(z) = a1z + a2z2 for which P5Cϕ iscomplex symmetric.In summary, it appears that there is no ϕ(z) = a1z + a1z2 for which PnCϕ is a complexsymmetric operator for all n. Moreover, we seem to lose a dimension in our solution set foreach larger value of n. It seems that by forcing each truncation to be in terms of the samecoefficients, we are limiting ourselves too far. In order to preserve complex symmetry fromPage 50 RHIT Undergrad. Math. J., Vol. 18, No. 1one truncation to the next, perhaps ϕ needs to introduce new coefficients into the corre-sponding matrix.Conjecture. If ϕ has a finite Taylor series, then there exists n such that PnCϕ is not acomplex symmetric operator.4 Upper Triangular Case, P2CϕFor ϕ(z) =∑∞k=0 akzk with a2 = 0, the matrix representation of P2Cϕ is upper triangular.1 a0 a020 a1 2a0a10 0 a12Figure 7: Matrix of P2Cϕ with a2 = 0If a0 = 0, the matrix above will be diagonal, and therefore symmetric. Other than thistrivial solution, there are no real values of a0 and a1 for which P2Cϕ is complex symmetricby the Strong Angle Test.5 Further QuestionsWe conclude with some questions for further research.1. For ϕ(z)=a1z, PnCϕ is a CSO for all n because the corresponding matrix will alwaysbe diagonal. Besides this trivial case, are there any symbols ϕ for which PnCϕ is aCSO for all n?2. Composition operators on H2 that are complex symmetric have yet to be completelyclassified, particularly in the difficult case when ϕ is not linear-fractional. Can trun-cated composition operators help solve this problem?3. What other properties beyond complex symmetry, meaningful in both finite and infinitedimensions, can be studied for truncated composition operators?References[1] Levon Balayan and Stephan Ramon Garcia. (2010). Unitary Equivalence to a ComplexSymmetric Matrix: Geometric Criteria. Operators and Matrices, 4(1): 53-76.[2] Stephen Ramon Garcia and Mihai Putinar. (2005). Complex Symmetric Operators andApplications. Transactions of the American Mathematical Society 358(3): 1285-1315.RHIT Undergrad. Math. J., Vol. 18, No. 1 Page 51[3] Carl Cowen and Barbara MacCluer. (1995). Composition Operators on Spaces of AnalyticFunctions. CRC Press, Inc.[4] Sivaram K. Narayan, Daniel Sievewright, and Derek Thompson (2016). Complex Symmet-ric Composition Operators on H2. Journal of Mathematical Analysis and Applications,443(1): 625-630.[5] James E. Tener. (2010). Unitary Equivalence to a Complex Symmertric Matrix. PomonaCollege, Claremont, CA.

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Complex Symmetry of Truncated Composition Operators

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