Vertical operators on the Bergman space on the upper half-plane

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1 Vertical operators on te Bergman space on te upper alf-plane Tis text is a roug draft. Objectives. Establis criteria of vertical operators (in oter words, operators tat are invariant under orizontal sifts) on te Bergman space on te upper alf-plane. Requirements. Bergman space of te analytic square integrable functions on te upper alf-plane, Berezin transform, sift operator, Fourier transform, Laplace transform. Bergman kernel on te upper alf-plane (sort review) Fact 1 (evaluation functional is bounded). Given a point w Π, denote by eval w te evaluation functional at w on te Bergman space A 2 (Π): It is known tat tis functional is bounded. eval w (f) := f(w). Definition 1 (Bergman kernel on te upper alf-plane). Given a point w Π, denote by K Π,w te function corresponding to te evaluation functional at te point w (tis function is called sometimes te Bergman kernel of te space A 2 (Π)): f A 2 (Π) eval w (f) = K Π,w, f. Fact 2 (explicit formula for te Bergman kernel on te upper alf-plane). Let w Π. It is known tat 1 K Π,w (ξ) = π(w ξ). 2 Exercise 1 (norm of te Bergman kernel on te upper alf-plane). Let w Π. Calculate te norm K Π,w 2 of te function K Π,w. Vertical operators, page 1 of 6

2 Berezin transform of bounded linear operators (sort review) Definition 2 (Berezin transform of a bounded linear operator). Let S : A 2 (Π) A 2 (Π) be a bounded linear operator. Te Berezin transform of S is defined by B(S)(w) := SK Π,w, K Π,w K Π,w, K Π,w. Exercise 2. Prove tat te Berezin transform of bounded linear operators is a linear map: B(S 1 + S 2 )(w) = B(λS)(w) = Exercise 3. Let S : A 2 (Π) A 2 (Π) be a bounded linear operator and S be its adjoint operator. Prove tat B(S ) = B(S). B(S )(w) = Exercise 4. Find a proof of te fact tat te Berezin transform of bounded linear operators is an injective map: if B(S) = 0, ten S = 0. Vertical operators, page 2 of 6

3 Multiplication operator on te real line Definition 3. Let g L (R). Define M g L(L 2 (R)) by (M g f)(x) := g(x)f(x). Exercise 5 (product of te multiplication operators). Let g 1, g 2 L (R). Calculate te product of te operators M g1 and M g2. Exercise 6 (multiplication operators commute). Let g 1, g 2 L (R). Prove tat M g1 M g2 = M g2 M g1. Definition 4. Given a number R denote by Θ te following function R C: Θ (x) := e i x. Fact 3 (criterion of multiplication operator on te real line). Let S L(L 2 (R)). Ten te following conditions are equivalent: (a) S is invariant under multiplication by Θ for all R: R SM Θ = M Θ S. (b) S is te multplication operator by a bounded function: m L (R) S = M m. Exercise 7. Prove tat (b) implies (a). Exercise 8. Find in te literature a proof tat (a) implies (b). reference to te proof. Study te proof. Write ere an exact Vertical operators, page 3 of 6

4 Multiplication operator on te positive alf-line Definition 5. Let σ L (R + ). Define M σ L(L 2 (R + )) by (M σ f)(x) := σ(x)f(x). Exercise 9 (product of te multiplication operators). Let σ 1, σ 2 L (R + ). Calculate te product of te operators M σ1 and M σ2. Definition 6. Given a number R denote by Θ + te restriction of Θ onto te positive alf-line: Θ + (x) := ei x (x R + ). Exercise 10 (criterion of multiplication operator on te positive alf-line). Let S L(L 2 (R + )). Prove tat te following conditions are equivalent: (a) S is invariant under multiplication by Θ + for all R: R SM Θ + = M Θ + S. (b) S is te multiplication operator by a bounded function: σ L (R + ) S = M σ. Vertical operators, page 4 of 6

5 Vertical operators on te positive alf-plane Definition 7 (orizontal sift operator). Let R. Define H : A 2 (Π) A 2 (Π) by (H f)(w) := f(w ). Exercise 11. Let R and w Π. Calculate H K Π,w. (H K Π,w )(z) = Te following fact is taken from papers of N. Vasilevski. Fact 4. Define R: A 2 (Π) L 2 (R + ) by R: A 2 (Π) L 2 (R + ), (Rφ)(x) = x 1 π φ(w) e i wx dµ(w), Tis operator R is an isometrical isomorpism, and its inverse R : L 2 (R + ) A 2 (Π) is given by: (R f)(z) = 1 ξf(ξ) e i zξ dξ. π R + Π Exercise 12. Let R and f L 2 (R + ). Prove tat R (Θ + f) = H R (f). (1) Vertical operators, page 5 of 6

6 Exercise 13. Let S L(A 2 (Π)). Prove tat te following conditions are equivalent: (a) S is invariant under orizontal sifts: R SH = H S. (b) RSR is invariant under multiplication by Θ + for all R: R RSR M Θ + (c) Tere exists a function σ L (R + ) suc tat S = R M σ R. (d) Te Berezin transform of S is a vertical function: = M Θ + RSR. z Π B(S)(z) = B(S)(I(z)). Vertical operators, page 6 of 6