The core operator and the compressed multipliers on $$M_{\psi ,\varphi }$$ -type submodules
Tóm tắt
Structure of submodules in
$$H^{2}({\mathbb {D}}^{2})$$
is very complicated. A good understanding of some special examples will shed light on the general picture. This paper studies the submodule
$$M_{\psi ,\varphi }=[\psi (z)-\varphi (w)]$$
and the corresponding quotient module
$$N_{\psi ,\varphi }=H^{2}({\mathbb {D}}^{2})\ominus M_{\psi ,\varphi }$$
. This paper calculates the Hilbert–Schmidt norm of the core operator and the commutators
$$[S^{*}_{z},S_{z}]$$
,
$$[S^{*}_{z},S_{w}]$$
and
$$[S^{*}_{w},S_{w}]$$
. This paper also studies the Fredholmness, the spectra, and the essential spectra of the compression operators
$$S_{z}$$
and
$$S_{w}$$
.
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