Probing deformed Kiselev-AdS Black Holes Surrounded by a Cloud of Strings through Thermodynamics, Radiative Features, and Shadow Observables

We investigate the thermodynamic, radiative, and optical properties of a deformed Schwarzschild--Anti-de Sitter black hole surrounded by a cloud of strings and immersed in a quintessence field. The model is described by a static and spherically symmetric metric whose lapse function contains the string-cloud parameter $\gamma$, the quintessence parameters $(\mathcal N,w)$, the cosmological constant $\Lambda$, and the deformation parameters $(\alpha,\beta)$. We derive the horizon thermodynamics in the extended phase space, including the Hawking temperature, thermodynamic volume, generalized first law, and Smarr relation. We also analyze the sparsity of Hawking radiation by using the geometric-optics absorption cross section determined by the critical impact parameter of null geodesics. The photon sphere and black-hole shadow are then studied numerically, showing how the deformation, string cloud, and quintessence contributions modify the characteristic optical radii. In the special limit $\beta\to0$ with $\alpha=Q^2$, the metric reduces to a charged-like AdS geometry, so the deformation parameter plays the role of an effective charge squared. Finally, we compare the theoretical shadow radius with the Event Horizon Telescope constraints for M87$^*$ and Sgr A$^*$, emphasizing that a complete constraint analysis requires a multidimensional scan of the parameter space.