Approximate spectral cosynthesis in the harmonically weighted Dirichlet spaces

Yıl 2023, Cilt: 52 Sayı: 3, 721 - 728, 30.05.2023

Faruk Yılmaz

https://doi.org/10.15672/hujms.1171901

Öz

For a finite positive Borel measure $\mu$ on the unit circle, let $\mathcal{D}(\mu)$ be the associated harmonically weighted Dirichlet space. A shift invariant subspace $\mathcal{M}$ recognizes strong approximate spectral cosynthesis if there exists a sequence of shift invariant subspaces $\mathcal{M}_k$, with finite codimension, such that the orthogonal projections onto $\mathcal{M}_k$ converge in the strong operator topology to the orthogonal projection onto $\mathcal{M}$. If $\mu$ is a finite sum of atoms, then we show that shift invariant subspaces of $\mathcal{D}(\mu)$ admit strong approximate spectral cosynthesis.

Anahtar Kelimeler

weighted Dirichlet spaces, invariant subspaces, strong approximate spectral cosynthesis

Kaynakça

• [1] A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81(17), 1948.
• [2] A. Borichev, D. Hadwin, and H. Yousefi, Stable and norm-stable invariant subspaces, J. Operator Theory 69(1), 3–16, 2013.
• [3] L. Brown and A. L. Shields, Cyclic vectors in the Dirichlet space, Trans. Amer. Math. Soc. 285(1), 269–303, 1984.
• [4] J. B Conway, A Course in Functional Analysis, Grad. Texts in Math. 96, Springer Science & Business Media, 2013.
• [5] J. B. Conway and D. Hadwin, Stable invariant subspaces for operators on Hilbert space, Ann. Polon. Math. 66, 49–61, 1997. Volume dedicated to the memory of Wlodzimierz Mlak.
• [6] O. El-Fallah, Y. Elmadani, and K. Kellay, Cyclicity and invariant subspaces in Dirichlet spaces, J. Funct. Anal. 270(9), 3262–3279, 2016.
• [7] O. El-Fallah, K. Kellay, J. Mashreghi, and T. Ransford, A Primer on the Dirichlet Space, Volume 203 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2014.
• [8] O. El-Fallah, K. Kellay, and T. Ransford, Cyclicity in the Dirichlet space, Ark. Mat. 44(1), 61–86, 2006.
• [9] O. El-Fallah, K. Kellay, and T. Ransford, On the Brown-Shields conjecture for cyclicity in the Dirichlet space, Adv. Math. 222(6), 2196–2214, 2009.
• [10] Y. Elmadani and I. Labghail, Cyclicity in Dirichlet spaces, Canad. Math. Bull. 62(2), 247–257, 2019.
• [11] M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes 19, de Gruyter Studies in Mathematics 1991.
• [12] D. Guillot, Fine boundary behavior and invariant subspaces of harmonically weighted Dirichlet spaces, Complex Anal. Oper. Theory 6(6), 1211–1230, 2012.
• [13] P. R. Halmos, A Hilbert Space Problem Book Volume 19 of Grad. Texts in Math. Springer-Verlag, New York-Berlin, second edition, 1982. Encyclopedia of Mathematics and its Applications, 17.
• [14] S. Luo and S. Richter, Hankel operators and invariant subspaces of the Dirichlet space, J. Lond. Math. Soc.(2) 91(2), 423–438, 2015.
• [15] N. K. Nikolski, Operators, Functions, and Systems: An Easy Reading Vol. 1 volume 92 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2002. Hardy, Hankel, and Toeplitz, Translated from the French by Andreas Hartmann.
• [16] N. K. Nikolski, Two problems on spectral synthesis, Journal of Soviet Mathematics 26(5), 2185–2186, 1984.
• [17] S. Richter, Invariant subspaces of the Dirichlet shift, J. Reine Angew. Math. 386, 205–220, 1988.
• [18] S. Richter, A representation theorem for cyclic analytic two-isometries, Trans. Amer. Math. Soc. 328(1), 325–349, 1991.
• [19] S. Richter and C. Sundberg, A formula for the local Dirichlet integral, Michigan Math. J. 38(3), 355–379, 1991.
• [20] S. Richter and C. Sundberg, Multipliers and invariant subspaces in the Dirichlet space, J. Operator Theory 28(1), 167–186, 1992.
• [21] S. Richter and C. Sundberg, Invariant subspaces of the Dirichlet shift and pseudocontinuations, Trans. Amer. Math. Soc. 341(2), 863–879, 1994.
• [22] S. Richter and F. Yilmaz, Regularity for generators of invariant subspaces of the Dirichlet shift, J. Funct. Anal. 277(7), 2117–2132, 2019.
• [23] W. Rudin, Function Theory in Polydiscs, W. A. Benjamin, Inc., New York- Amsterdam, 1969.
• [24] S. M. Shimorin, Approximate spectral synthesis in the Bergman space, Duke Math. J. 101(1), 1–39, 2000.
• [25] F. Yilmaz, Approximation of invariant subspaces in some Dirichlet-type spaces, Complex Anal. Oper. Theory 12(8), 1959–1972, 2018.