# Hilbert Space Harmonic Functions Homework

## ANNOUNCEMENTS

- Beginning of instruction for the Spring term is Monday, March 30, 2015.

## COURSE DESCRIPTION

This course presents some maximal and ergodic theorems and results about harmonic functions and Hardy spaces from harmonic analysis, and it is an introduction to some propertis of compact operators on a Hilbert space and the theory of commutative \(C^*\)-algebras.

## PREREQUISITES

Ma 108 or previous exposure to metric space topology and Lebesgue measure.

## SCHEDULE

Monday-Wednesday-Friday, 11:00-11:55 am, Sloan 257.

## OFFICE HOURS

Monday-Wednesday-Friday, 12:00-12:40 pm, Sloan 358, and by appointment with the TA of the course.

## POLICIES

The evaluation will be based on three to four sets of homework and class participation.

## TOPICS COVERED

- Harmonic analysis: maximal functions and the Hardy-Littlewood maximal theorem, the maximal and Birkoff ergodic theorems, harmonic and subharmonic functions, theory of \(H^p\)-spaces and boundary values of analytic functions.
- Operator theory: compact operators, trace and determinant on a Hilbert space, orthogonal polynomials, the spectral theorem for bounded operators, Banach algebras and the theory of commutative \(C^*\)-algebras.

## OUTLINES OF LECTURES

Date | Outline of the Lecture |
---|---|

Monday, March 30th | The Hardy-Littlewood maximal inequality was proved and it was used to prove the Lebesgue differentiation theorem. |

Wednesday, April 1st | We had a brief discussion about the sharpness of the constants in the Hardy-Littlewood maximal inequality and the \( L^p \)-norm of the maximal function. In this direction, one can read more about Stein's spherical maximal theorem. Then, measure preserving transformations were introduced and several examples of such trnasformations were given. The mean ergodic theorem, and the maximal ergodic theorem for the counting measure on \( \mathbb{Z} \) endowed with translation, was proved. The idea of the proof of the maximal ergodic theorem for general measure spaces was given, which will be completed in the next lecture. |

Friday, April 3rd | The maximal ergodic theorem for a general measure space \( (X, \mathcal{M}, \mu) \), endowed with a measure preserving transformation \( \tau: X \to X \), was proved. Then, for the case \( \mu(X) < \infty \), the pointwise ergodic theorem was proved. That is, it was shown that for any \( f \in L^1(X, \mu) \), the sequence \( \frac{1}{m} \sum_{k=0}^{m-1} f( \tau^k(x)) \) converges for almost every \( x\) in \( X\), as \( m \to \infty \). |

Monday, April 6th | As a corollary of the mean ergodic theorem and the pointwise ergodic theorem, it was shown that if \( f \in L^1(X, \mu) \) then for any measure preserving transformation \( \tau : X \to X \), the \( L^1 \)-norm of \( \left ( P'f \right ) (x) := \lim_m \frac{1}{m}\sum_{k=0}^{m-1} f( \tau^k(x)) \) is bounded by \( ||f||_{L^1} \). Then the notion of an ergodic measure preserving transformation was introduced and the Birkhoff ergodic theorem was proved: if \(f \in L^1(X, \mu), \mu(X)=1, \) and \( \tau \) is ergodic, then \(\lim_m \frac{1}{m}\sum_{k=0}^{m-1} f( \tau^k(x)) = \int_X f \,d\mu \) for almost every \(x \) in \( X\). At the end, the ergodicity of irrational rotaions on a circle of length 1 was discussed. |

Wednesday, April 8th | Two examples were studied in detail: irrational rotations on the circle of length 1 and the doubling map on \( (0, 1] \). Using the equidistribution theorem it was shown that the irrational rotation \( \tau: x \to x + \alpha \) on \( \mathbb{S}^1=\mathbb{R}/ \mathbb{Z} \) with the Lebesgue measure, for \( \alpha \in \mathbb{R} \setminus \mathbb{Q} \) fixed, is ergodic. Moreover, it was shown that this system is uniquely ergodic in the sense that the Lebesgue measure is the unique \( \tau\)-invariant Borel probability measure on \( \mathbb{S}^1 \). For the doubling map, it was shown that the system satisfies a stronger condition, namely the mixing condition, i. e., for \( \tau: x \to 2 x \) mod 1 on \( (0, 1] \), any measurable subsets \( E, F\) are asymptotically independent, in the sense that \( \lim_n m (\tau^{-n}(E) \cap F) = m(E) m(F)\). At the end it was explained by an easy example that the latter is not uniquely ergodic. There are more examples on this page. |

Friday, April 10th | The distribution function \(m_f \) of a measurable function on a measure space \( (X, \mathcal{M}, \mu)\) was defined: \( m_f(t) = \mu (\{x \in X; |f(x)| >t \}), t>0 \). After observing that \(m_f \) is right continuous and monotone, its discontinuities were analysed. Then for a given nondecreasing function \(F \) on \( [0, \infty ) \) with \( \lim_{t \to 0^+} F(t) = F(0) =0 \), a measure \( \nu \) was defined by \( \nu([0,t))=F(t)\), and it was shown that \( \int_X F(|f(x)|) d \mu(x) = \int_{t \geq 0} m_f(t) dt. \) The weak \( L^p\)-space \(L^p_w(X, \mu) \) was introduced using the weak \( L^p \)-norm \( ||f||_{p,w}^p = \sup_{t >0} (t^p m_f(t))\) and it was shown that this norm is dominated by the \(L^p \)-norm. It was also proved that if \( 1 \leq p_1 < p < p_2 < \infty \), then \( L^{p_1}_w (X, \mu) \cap L^{p_2}_w (X, \mu) \subset L^p(X, \mu) \), and an upper bound for the \( L^p \)-norm in terms of the weak \( L^{p_i} \)-norms was derived. At the end the Marcinkiewicz interpolation theorem was stated, which will be proved in the next lecture. |

Monday, April 13th | The Marcinkiewicz interpolation theorem was proved and as its corollary it was shown that for any \( 1 < p \) there is a constant \( C_{p, d} \) such that \( ||f^*||_{L^p(\mathbb{R}^d)} \leq C_{p, d} ||f||_{L^p(\mathbb{R}^d)} \) for any \( f \in L^p(\mathbb{R}^d) \), where \(f^* \) denotes the Hardy-Littlewood maximal funtion of \(f \). For more applications of the theorem, one can read about how it is used to prove the boundedness of the Hilbert transform. In this direction, one can read the paper, Recent progress in classical Fourier analysis, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), Vol. 1, pp. 95-118. Canad. Math. Congress, Montreal, Que., 1975, by Charles Fefferman. At the end, it was observed by an example that the weak \( L^p \)-norm is not a norm for any \( p\geq 1 \): it does not satisfy the triangle inequality. |

Wednesday, April 15th | We had a brief discussion about the Hilbert transofrm and applications of the Marcinkiewicz interpolation theorem. Then, by proving an inequality for the weak \(L^p \)-norm, which is weaker than the triangle inequality, it was shown that the weak \( L^p \)-space is a vector space. For \( p>1 \), the Calderon norm on \( L^p_w \) was introduced and its important properties were discussed. At the end, basic tools for solving the Dirchlet probelm for the unit disk were reviewed (Fourier series, Dirichlet kernel, Poisson kernel, convolution product and its properties). |

Friday, April 17th | First we reviewed basic statements about pointwise convergence of the Fourier series of a function on the circle to the function. Then the notion of a family of good kernels was introduced and it was shown any such family provides an approximation of identity for the convolution product. Then it was discussed that the family of Dirichlet kernels \( D_N(x) = \sum_{n=-N}^{N} e^{inx} \) is not a good kernel, however, its Cesaro means, namely the Fejer kernel given by \( F_N (x) = \frac{1}{N} \sum_{i=0}^{N-1} D_i(x) \) is a good kernel. Therefore the Fourier series of any continuous funtion on the circle is uniformly Cesaro summale to the funtion. Then the notions of Abel means and Abel summability were explained, and at the end it was shown that the Poisson kernel \( P_r (\theta) = \sum_{n \in \mathbb{Z}} r^{|n|} e^{in \theta} \) is a family of good kernels as \(r \) approaches \(1 \) from below. This provides the necessary tools for solving the Dirichlet porblem for the disk, which will be done in the next lecture. |

Monday, April 20th | The solution of the Dirichlet problem for the unit disk in terms of the Poisson kernel and the uniquess of the solution for a continuous boundary condition on the circle were worked out. That is, it was shown that given a continuous function \(f \) on the unit circle, the following function defined in the polar coordinates \( u(r, \theta) = (f * P_r ) (\theta) \), where \( P_r(\theta) = \frac{1-r^2}{1 -2 r \cos \theta + r^2}\) is the Poisson kernel, is the unique solution to the Laplace equation \( \Delta u = u_{rr}+ \frac{1}{r} u_r + \frac{1}{r^2} u_{\theta \theta} = 0 \) such that as \(r \) tends to 1 from below, \( u(r, \theta) \) converges to \(f(\theta) \) uniformly. |

Wednesday, April 22nd | Basic propoerties of the Fourier transform, the space of Schwartz functions, convolution product, and the Fourier inversion theorem were recalled. Then the Laplace equation in the upper half plane \( \Delta u= u_{xx} + u_{yy}=0 \) with a boundary condition \( u(x, 0) = f(x) \), where \(f \) is a Schwartz function, was considered. The equation was solved using the Fourier transform and along the way the Poisson kernel for the upper half plane, \(\mathcal{P}_y (x) = \frac{y}{\pi(x^2 + y^2)}, y>0, x \in \mathbb{R},\) was derived, which turns out to give a good family of kernels as \( y \to 0^+\) and solves the problem. |

Friday, April 24th | It was shown that given a Schwartz function \( f \) on \( \mathbb{R} \), the function \( u(x, y) = (f * \mathcal{P}_y)(x), \) where \(\mathcal{P}_y (x) = \frac{y}{\pi(x^2 + y^2)} \) is the Poisson kernel, satisfies \( \Delta u=0, \) \( u(x, y) \to f(x) \) uniformly as \( y \to 0^+, \) and it vanishes at infinity in the sense that \( \lim_{|x|+y \to \infty}u(x, y) =0.\) Then the mean value property for harmonic functions was proved and it was used to show that if \( u\) is a continous function on the closure of the upper halp plane, such that \(u(x, 0)=0, \) \( \Delta u =0, \) and \( u\) vanishes at \(\infty, \) then it is identically \( 0 \). At the end the maximum princple for harmonic functions proved, namely that, if \( u \) is continuous on the closure of a bounded open region \( \Omega \) and \( \Delta u=0 \) inside the region, then \( \text{max}_{\bar \Omega} |u| = \text{max}_{\partial \Omega} |u|,\) where \( \partial \Omega = \bar \Omega \setminus \Omega. \) |

Monday, April 27th | The notion of the Lebesgue set of a locally integrable function on \( \mathbb{R}^d\) was introduced and it was explained that the convolution \( f * K_\delta \) of any \( L^1\)-function \(f \) with an approximation to the identity \(K_\delta \) tends to \( f\) at any point of the Lebesge set of \( f\), therefore at almost every point. It was then proved that for any \( f\in L^1([- \pi, \pi]) \), \( \lim_{r\to 1^-} \sum_{n \in \mathbb{Z}}a_n |r|^n e^{inx} = f(x) \) for almost every \(x. \) In order to start the study of the boundary values of analytic functions, Fatou's theorem was proved. That is, any bounded holomorphic function on the disk has radial limits at almost every \( - \pi \leq \theta \leq \pi. \) Following this, the Hardy space of the disk \( H^2(\mathbb{D}) \) was introduced and its main properties were analyzed. |

Wednesday, April 29th | We had a brief discussion about the extension of the Fourier transform on Schartz functions to a unitary map on \( L^2(\mathbb{R})\). Then, the Hardy space \( H^2(\mathbb{R}^2_+) \) of the upper half-plane \( \mathbb{R}^2_+ \) was introduced, which consists of all analytic functions \( F \) in \( \mathbb{R}^2_+ \) such that \( ||F||_{H^2}^2 := \sup_{y>0} \int_\mathbb{R} |F(x+iy)|^2 \,dx \) is finite. As a typical example, it was shown that the function defined as \( F(z) = \int_\mathbb{R} {\hat F}_0 (\xi) e^{2 \pi i z \xi }, \Im (z) >0,\) where \( {\hat F}_0 \in L^2((0, \infty))\), is in the Hardy space of \( \mathbb{R}^2_+ \) and \(||F||_{H^2} = ||{\hat F}_0||_{L^2} \). Then it was stated that any element of the Hardy space is of this form, and in order to prove this assertion, it was shown that given \( F \in H^2(\mathbb{R}^2_+), \) it is bounded in any proper halp-plane \( \Im (z) \geq \delta > 0 \), and that \( {\hat F}_y (\xi) e^{2 \pi y \xi} := \int_\mathbb{R} e^{-2\pi i x \xi} F(x+i y)\, dx \, e^{2 \pi y \xi} \) is independent of \(y>0 \) for almost every \( \xi. \) |

Friday, May 1st | It was proved that any function \( F \) in the Hardy space of the upper half-plane can be written as \( F(z) = \int_\mathbb{R} {\hat F}_0 (\xi) e^{2 \pi i z \xi }, \Im (z) >0,\) for some \( {\hat F}_0 \in L^2((0, \infty))\). It was then shown that \(\lim_{y \to 0^+} F(x+iy) = F_0(x) \) exists for almost every \( x \) and the limit makes sense both as a limit in \( L^2\)-norm and as pointwise limit. The Fourier transofrm of \( F_0 \) was shown to be supported in \( (0, \infty) \) and equal to the function appearing above, which jusitfies the notation \( {\hat F}_0 \) used. |

Monday, May 4th | A few remarks were made. First, the boudary values of functions in the Hardy space of the upper half-plane form all functions \( f\) in \( L^2(\mathbb{R}) \) such that the Fourier transform \(\hat f \) is supported in \( (0, \infty). \) Let us denote this space by \( S\). The second remark was that, if we consider the orthogonal projection \( P: L^2(\mathbb{R}) \to S \), then \( \widehat {P(f)} = \chi_{(0, \infty)} \hat f \) for any \( f \in L^2(\mathbb{R}). \) It was then shown that if \( F \in H^2(\mathbb{R}^2_+)\) is the unique element with boundary values \( P(f) \), then \( F(z)= \frac{1}{2 \pi i} \int_{\mathbb{R}} \frac{f(t)}{t-z}\, dt, \Im(z) >0.\) In the third remark, it was explained that \(P \) is a Fourier multiplier operator and the Hilbert transform \( H, \) which is defined by \( P= \frac{1}{2}(1 + i H), \) was elaborated on. Then the focus was turned to weak solutions of partial differential equations of type \( Lu = f \), where \(L \) is a partial differential operator on \( \mathbb{R}^d \) with constant complex coefficients. Indeed, the main ideas in the proof of the existence of a weak solution for \( Lu = f,\) where \( f \in L^2(\Omega) \) and \(\Omega \) is a bounded open subset of \( \mathbb{R}^d,\) were outlined. |

Wednesday, May 6th | The notion of a Banach algebra over complex numbers was introduced and after reviewing several examples of such algebras, their basic propoerties were analysed. In particular the spetrum \( \sigma(a) \) of an element \(a\) in a unital algebra was introduced to be the set \( \{ \lambda \in \mathbb{C};\, \lambda - a \notin \text{Inv}(A)\} \) and it was shown that for any polynomial \( p \) with complex coefficients \( \sigma(p(a))= p(\sigma (a)).\) At the end, it was shown that if \( ||a|| < 1 \) in a unital Banach algebra \( A \), then \( 1-a \) is invetible in \( A\) and its inverse is given by the Neumann series \( (1-a)^{-1} = \sum_{n=0}^\infty a^n.\) |

Friday, May 8th | Let \( A \) be a unital Banach algebra. It was shown that the set of invertible elements in \( A, \text{Inv}(A),\) is an open subset and the map \( f: \text{Inv}(A) \to A, f(a) := a^{-1} \) is differentiable at any invertible \(a\) and \( (f'(a))(b) = - a^{-1}ba^{-1}. \) Then it was proved that for any \( a \in A,\) the spectrum \( \sigma(a)\) is a compact subset contained in the disc of radius \( ||a||\) centered at the origin. Gelfand's theorem about the nonemptiness of the spectrum of any element in a unital Banach algebra was proved, and it was used to show that if every non-zero element of a Banach algebra is invertible then the Banach algebra is isomorphich to complex numbers (Gelfand-Mazur theorem). At the end, the spectral radius \(r(a):=\sup_{\lambda \in \sigma(a)} |\lambda| \) was introduced and the Beurling theorem was stated and its proof was elaborated on, which will be continued in the next lecture. The theorem states that \( r(a) = \inf_n ||a^n||^{1/n}= \lim_{n} ||a^n||^{1/n},\) for any element \( a\) of a unital Banach algebra. |

Monday, May 11th | The proof of the Beurling theorem was completed. Then the unitization of a non-unital Banach algebra was discussed. It was shown that the closure of any modular ideal in a Banach algebra is also modular, which implies that if the ideal is maximal then it is closed. The set of characters \( \Omega(A) \) of a Banach algebra was introduced to be the set of all non-zero algebra homomorphisms from \( A\) to \( \mathbb{C}. \) It was shown that if the Banach algebra \( A\) is unital and commutative then any \( \tau \in \Omega(A)\) is continuous with \( ||\tau||=1 \), and the map that sends any \( \tau \in \Omega(A) \) to \(\text{Ker}(\tau) \) is a bijection between the set of characters and the set of maximal ideals of \( A\), which in particular implies that \( \Omega(A)\) is not empty. |

Wednesday, May 13th | Let \( A\) be a commutative Banach algebra. It was shown that if \( A \) is unital then \( \sigma(a) = \{\tau(a); \, \tau \in \Omega(A) \}, \) and if \( A\) is non-unital then \( \sigma(a) = \{\tau(a); \, \tau \in \Omega(A) \} \cup \{ 0\},\) for any \( a \in A, \) where \( \sigma(a)\) denotes the spectrum of \( a \) and \( \Omega(A) \) is the space of characters of \( A\). In the non-unital case, \( \sigma(a)\) is defined to be the specrum of \( a \) in the unitization of the Banach algebra. This implies that \( \Omega(A) \) is contained in the closed unit disk \( S \) of the dual space \( A^*\). The weak* topology on \( A^* \) and the Banach-Alaoglu theorem were explined, which states that \( S \) is weak* compact. The set \( \Omega(A) \) endowed with the weak* topolpogy is called the spectrum of \( A\). It was explained that \( \Omega(A) \) is a locally compact Hausdorff space if \( A \) is non-unital, and it is a compact Hausdorff space if \( A \) is unital. The Gelfand transform was introduced which sends any \( a \in A\) to \( \hat a \in C_0(\Omega (A)) \) defined by \( \hat a (\tau) = \tau(a) \). At the end, the Gelfand representation theorem was proved. The latter asserts that the Gelfand transform defines a norm decreasing map from \( A \) to \( C_0(\Omega(A)) \), i.e. \(||\hat a ||_\infty \leq ||a||, \) for any \( a \in A;\) moreover, if \( A \) is unital then \( \sigma(a) = \hat a (\Omega(A)),\) and if \( A\) is non-unital then \( \sigma(a)= \hat a (\Omega(A)) \cup \{ 0\} .\) |

Friday, May 15th | It was shown that if \(A \) is a unital Banach algebra generated by \(1\) and an element \(a\), then the Gelfand transform of \(a\), \(\hat a : \Omega(A) \to \sigma(a) \) defined by \(\hat a (\tau) = \tau(a), \) is a homeomorphism of the topological spaces \( \Omega(A) \) and \( \sigma(a)\). Then the notion of a \(C^*\)-algebra was defined to be a complete Banacah *-algebra with a submultiplicative norm such that \(||a^* a ||=||a||^2 \) for any element \( a \) of the algebra. After giving examples of \( C^*\)-algebras and introducing the notion of a unitary element, a prjection, a self-adjoint element, and a normal element of a \(C^*\)-algebra, and discussing their basic properties, we proved some general results. That is, we showed that if \(a \) is a self-adjoint element of a \( C^*\)-algebra \(A,\) then \( r(a)=||a|| \), and concluded that the norm on \(A \) is the unique norm that makes it a \(C^*\)-algebra. Then we proved that any *-homomorphism from a unital Banach *-algebra to a \(C^*\)-algebra is necessarily norm-decreasing. Also we showed that if \( a\) is a self-adjoint element of a \( C^*\)-algebra, then \(\sigma(a) \subset \mathbb{R}. \) We remarked that the last two statements hold even if we remove the unital condition, which will be justified in the next lecture after introducing the muliplier algebra of a \(C^*\)-algebra. |

Monday, May 18th | The multiplier algebra \(M(A)\) of a \( C^*\)-algebra \( A\) was introduced to be the set of all double centralizers of \( A\), where by defintion a double centralizer is a pair \((L, R) \) of bounded linear maps on \(A \) satisfying the relations \( L(ab)=L(a)b, R(ab)=a R(b), R(a)b=aL(b), \) for any \(a, b \in A.\) After defining a product, an involution, and a norm on \( M(A) \), it was shown that it forms a \(C^* \)-algebra, in which \(A \) embeds isometrically as an ideal. It was then shown that there is a (necessarily unique) norm on the unitization of \( A\) that makes it a \( C^*\)-algebra. After remarking that the charater space of any commutative \( C^*\)-algebra is non-empty, the theorem of Gelfand was proved: If \( A\) is a (non-zero) commutative \( C^*\)-algebra then the Gelfand representaion sending any \( a \in A\) to \(\hat a,\) where \( \hat a (\tau) = \tau (a) \) for any \(\tau \in \Omega(A), \) is an isometric *-isomorphism between \( A\) and \( C_0(\Omega(A)). \) |

Wednesday, May 20th | It was shown that if \( a\) is a normal element in a unital \(C^*\)-algebra \( A\), then there is a unique unital *-homomorphism \( \varphi : C(\sigma(a)) \to A \) such that \( \varphi(z) = a, \) where \(z : \sigma(a) \to \mathbb{C} \) is the inclusion map. The homomorphism \( \varphi \) is called the functional calculus at \(a \in A \), and since \( \varphi(p) = p(a) \) for any polynimal \( p \), it is convenient to use the notation \( f(a) = \varphi(f) \) for any \( f \in C(\sigma(a)). \) Then the spectral mapping theroem was proved, which asserts that \( \sigma(f(a)) = f( \sigma(a)), \) and that if \( g \in C(\sigma(f(a))), \) then \( (g \circ f)(a) = g(f(a)). \) Then our focus was turned to the spectral theory of bounded operators on a Hilbert space. Basic properties of a compact operator and a normal operator on a Hilbert sapce were reviewed, the spectral theorem for the compact normal operators on a Hilbert space was stated, and the idea of its proof was elaborated on. |

Wednesday, May 27th | The spectral theorem for normal compact operators on a Hilbert space was proved. Then the singular values \( s_n (T) \) of a compact operator on a Hilbert space were defined to be the eigenvalues of \( |T| = (T^*T)^{1/2} \) ordered decreasingly: \( s_1(T) \geq s_2(T) \geq s_3(T) \cdots \to 0 .\) We defined the trace class operators on a Hilbert space to be the set of compact operators \( T \) such that \(||T||_1 := \sum_{n=1}^\infty s_n(T) < \infty. \) In order to prove the main properties of the trace class operators, we proved the following statements. First, we showed that if \(T \) is a compact operator, then there are orthonormal sequences \( \{e_n\}, \{ f_n\} \) in the Hilbert space such that \(Tx = \sum_n s_n(T)(x, e_n) f_n\) for any vector \( x\). Second, it was shown that if \(T \) is a trace class operator and \(\{x_m\}, \{ y_m\} \) are orthonormal sequences in the Hilbert space, the \( \sum_m |(Tx_m, y_m)| \leq ||T||_1. \) It was then shown that given a trace class operator \(T \) and an orthonormal basis \( \{x_m \},\) the trace defined by \( \text{Tr}(T) = \sum_m (Tx_m, x_m) \) is well defined and independent of the choice of the orthonormal basis. It was also proved that if \( A\) and \(B\) are bounded operators on a Hilbert space such that \( AB\) and \( BA\) are trace class operators, then \( \text{Tr}(BA)= \text{Tr}(AB). \) At the end, given a compact Hausdorff space \( \Omega\) and a Hilbert space \( H \), the notion of a spectral measure relative to \( (\Omega, H) \) was introduced. |

Friday, May 29th | A few properties of a sequilinear form on a Hilbert space were reviewed and an example of a spectral measure was given. It was explained that if \(E \) is a spectral measure relative to \( (\Omega, H)\), where \(\Omega \) is a compact Hausdorff space and \(H \) is a Hilbert space, then for a given bounded Borel measurable function \(f \) on \( \Omega\), the map \(\sigma_f: H \times H \to \mathbb{C} \) defined by \( \sigma_f( (x, y) ) = \int f \, dE_{x, y} \) is a bounded sesquilinear form on \( H \); here \( E_{x,y}(S) = (E(S)x, y) \) for any Borel set \( S \). Thus, one can define \( u = \int f \,dE \) to be the unique bounded operator on \(H \) such that \( (u(x), y) = \int f \,dE_{x,y}\) for all \(x, y \in H. \) It was then discussed that if \( \varphi : C(\Omega) \to B(H) \) is a unital *-homomorphism, then there is a unique spectral measuer \( E \) relative to \( (\Omega, H)\) such that \(\varphi(f)=\int f \, dE \) for any \(f \in C(\Omega). \) Combining the latter with the functional calculus for normal elements of \(C^*\)-algebras, the spectral theorem was proved: if \( u \) is a normal operator on a Hilbert space \(H \), then there is a unique spectral measure relative to \((\sigma(u), H) \) such that \( u = \int z \, dE,\) where the map \( z \) is the inclusion of \( \sigma(u)\) in \(\mathbb{C}. \) |

## HOMEWORK

Distribution Date | Due Date | Homework | Solutions |
---|---|---|---|

Monday, April 13th | Monday, April 27th | Homework 1 | |

Monday, April 27th | Friday, May 8th | Homework 2 | |

Friday, May 8th | Friday, May 22nd | Homework 3 | |

Friday, May 22nd | Wednesday, June 3rd | Homework 4 |

## EXAMS

There are no exams for this course.

## REFERENCES

- Barry Simon, A Course in Analysis, Part 3: Harmonic Analysis, draft.
- Barry Simon, A Course in Analysis, Part 4: Operator Theory, draft.
- Gerard J. Murphy, \(C^*\)-algebras and operator theory, Academic Press, 1990.
- Elias M. Stein, Rami Shakarchi, Real analysis. Measure theory, integration, and Hilbert spaces. Princeton Lectures in Analysis, III, Princeton University Press, 2005.

Calculus and Analysis > Harmonic Analysis > Harmonic Functions >

MathWorld Contributors > Rowland, Todd >

## Harmonic Conjugate Function

The harmonic conjugate to a given function is a function such that

is complex differentiable (i.e., satisfies the Cauchy-Riemann equations). It is given by

where , , and is a constant of integration.

Note that is a closed form since is harmonic, . The line integral is well-defined on a simply connected domain because it is closed. However, on a domain which is not simply connected (such as the punctured disk), the harmonic conjugate may not exist.

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